quantum teleportation and multi-photon entanglement · realisierung von quantenteleportation,...

Quantum Teleportation andMulti-photon Entanglement

Dissertation zur Erlangung des Grades einesDoktors der Naturwissenschaften

eingereicht von

M.Sc. Jian-Wei PanUniversity of Science and Technology of China

Durchgefuhrt am Institut fur Experimentalphysikder Universitat Wien

beio.Univ.Prof.Dr. Anton Zeilinger

Gefordert vom Fonds zur Forderung der wissenschaftlichen Forschung, Projekte

S6502 und F1506 und durch das TMR-Netzwerk The Physics of Quantum

Information der Europaischen Kommission.

Contents

1 Introduction 5

2 Manipulation of Entangled States 10

2.1 Quantum network and its applications . . . . . . . . . . . . . 11

2.2 Practical schemes for entangled-state analysis . . . . . . . . . 15

2.2.1 Bell-state analysis . . . . . . . . . . . . . . . . . . . . . 16

2.2.2 GHZ-state analyzer . . . . . . . . . . . . . . . . . . . . 20

2.3 Polarization-entangled photon pairs . . . . . . . . . . . . . . . 29

3 Quantum Teleportation 33

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 Quantum teleportation–the idea . . . . . . . . . . . . . . . . . 34

3.2.1 The problem . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2.2 The concept of quantum teleportation . . . . . . . . . 35

3.3 Experimental teleportation . . . . . . . . . . . . . . . . . . . . 40

3.3.1 Experimental scheme . . . . . . . . . . . . . . . . . . . 40

3.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4 Entanglement Swapping 53

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2 Theoretical scheme . . . . . . . . . . . . . . . . . . . . . . . . 54

i

ii CONTENTS

4.3 Experimental entanglement swapping . . . . . . . . . . . . . . 56

4.4 Generalization and applications . . . . . . . . . . . . . . . . . 61

5 Three-photon GHZ entanglement 64

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.2 Experimental Set-up . . . . . . . . . . . . . . . . . . . . . . . 65

5.3 Observation of three-photon entanglement . . . . . . . . . . . 72

5.4 Discussion and conclusion . . . . . . . . . . . . . . . . . . . . 75

6 Experimental tests of the GHZ theorem 76

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.2 The conflict with local realism . . . . . . . . . . . . . . . . . . 77

6.2.1 GHZ theorem . . . . . . . . . . . . . . . . . . . . . . . 77

6.2.2 Generalization to conditional GHZ state . . . . . . . . 81

6.3 Experimental results . . . . . . . . . . . . . . . . . . . . . . . 84

6.4 Discussion and Prospects . . . . . . . . . . . . . . . . . . . . . 90

7 Conclusions and outlook 92

Zusammenfassung

Die vorliegende Dissertation ist das Ergebnis theoretischer und experimentellerArbeiten uber die Physik von Mehrteilcheninterferenz. Die theoretischenErgebnisse zeigen, daß man Quantenverschrankung mit einem Quantennet-zwerk aus einfachen Quantenlogikgattern und einer kleinen Anzahl von Qubitskontrollieren und manipulieren kann. Da es bis jetzt keine experimentelleDurchfuhrung von Quantengattern fur zwei unabhangig erzeugte Photonengibt, prasentieren wir hier eine realisierbare Methode, verschrankte Viel-teilchenzustande zu erzeugen und zu identifizieren.

In der experimentellen Arbeit wurden die zum Studium von neuarti-gen Vielteilcheninterferenzphanomenen notigen Techniken von Grund auf en-twickelt. Wir berichten in dieser Arbeit uber die erstmalige experimentelleRealisierung von Quantenteleportation, ’Entanglement Swapping’ und derErzeugung von Dreiteilchenverschrankung mithilfe einer gepulsten Quellefur polarisationsverschrankte Photonen. Mit der Quelle fur Dreiteilchen-verschrankung wurde das erste Experiment zum Test von lokalrealistischenTheorien ohne Ungleichungen durchgefuhrt.

Die in diesen Experimenten entwickelten Methoden sind von großer Be-deutung fur Forschungen auf dem Gebiet der Quanteninformation und furzukunftige fundamentale Experimente der Quantenmechanik.

1

Abstract

The present thesis is the result of theoretical and experimental work on thephysics of multiparticle interference. The theoretical results show that aquantum network with simple quantum logic gates and a handful of qubitsenables one to control and manipulate quantum entanglement. Because of thepresent absence of quantum gate for two independently produced photons,in the mean time we also present a practical way to generate and identifymultiparticle entangled state.

The experimental work has thoroughly developed the necessary tech-niques to study novel multiparticle interference phenomena. By making useof the pulsed source for polarization entangled photon pairs, in this thesis wereport for the first time the experimental realization of quantum teleporta-tion, of entanglement swapping and of production of three-particle entangle-ment. Using the three-particle entanglement source, here we also present thefirst experimental realization of a test of local realism without inequalities.

The methods developed in these experiments are of great significance bothfor exploring the field of quantum information and for future experiments onthe fundamental tests of quantum mechanics.

2

Acknowledgements

I am indebted to my advisor, Professor Anton Zeilinger, for his guidanceand support throughout the course of my work leading to this thesis. Hetaught me with wisdom, encouragement, rich knowledge, insight and a deepunderstanding of physics, and more importantly the way to conduct scientificresearch. I am very grateful that he has been always available to discuss themany problems and questions I brought him. I would also like to thank himfor critically reviewing this thesis.

I am very grateful to my second advisor, Professor Helmut Rauch, whowarmly recommended and supported my application for the Austrian Chan-cellor Fellowship from the Austrian Academic Exchange Service, which en-abled my continuation of physics study in Austria.

I would like to express my deep appreciation to my former and presentcolleagues in the Quantum Optics and Foundations of Physics research group.Special thanks go to my friend and permanent colleague, Dr. Dik Bouwmeester,who introduced me to the subject of quantum optics and taught me manydetails of the experiment; he also impressed me with his devotion and ini-tiative. I also especially thank Professor Harald Weinfurter and MatthewDaniell, with whom I collaborated on most of the work. Matthew also hascarefully read through some of the chapters in this thesis. Thanks also tothe other colleagues in the photon laboratory, Dr. Birgit Dopfer, Dr. KlausMattle, Dr. Markus Michler, Dr. Michael Reck, Dr. Surasak Chiangga,Dr. Gregor Weihs, Thomas Jennewein, Alois Mair, Markus Oberparleitnerand Christoph Simon. To the people in the atom laboratory, Professor JorgSchmiedmayer, Dr. Markus Arndt, Dr. Stefan Bernet, Dr. Johannes Den-schlag, Dr. Sonja Frank, Donatella Cassettari, Claudia Keller, Olaf Nairz,and Gerbrand von der Zouw, whose instruments I sometimes stole. And toMrs. Christine Obmascher, Professor Zeilinger’s secretary, for her quiet effi-ciency in the office, and for her continuous help on various matters throughout

3

the years.

Many helpful discussions and much of my knowledge about Bell’s in-equalities are due to my friend and colleague Professor Marek Zukowski, thepermanent visitor to our group.

Dr. Ramon Risco Delgado and Bjorn Hessmo are always remembered. Ienjoyed our discussions about physics, the meaning of life, and all the rest,especially the delicious fish cooked by Bjorn.

This work would have been impossible without the patience and under-standing of my wife Xiao-qing, who supported me during the ups and downsthat are inevitable in such a major undertaking. My parents have providedme with invaluable support through my entire life and education. They haveencouraged and supported me both for starting my undergraduate study ina distant city, and for continuing my doctorate study in another country faraway from my homeland.

Among the many very good teachers I met throughout my academic ca-reer, I especially thank Professor Yong-de Zhang, my undergraduate andgraduate advisor, for his outstanding guidance and continuous concern aboutmy career.

I gratefully acknowledge the support of the Austrian Academic ExchangeService during my study. The financial support of the research in this thesiswas partially from the Austrian Fonds zur Forderung der WissenschaftlichenForschung who with the Schwerpunkt Quantenoptik (Project No. S06502),Project No. F1506 and the TMR-Network ”The Physics of Quantum Infor-mation” of the European Commission.

The attentive reader might notice that a number of text paragraphs weretaken from joint papers of our group because the formulations found thereare difficult to improve.

Finally I sincerely thank all my friends, from all over the world, who mademy years in Innsbruck and Vienna so delightful.

Chapter 1

Introduction

Superposition, one of the most distinct features of the quantum theory, has

been demonstrated in numerous particle analogs of Young’s classic double-

slit interference experiment, such as in electron interferometer [Marton et al.,

1954], neutron interferometer [Rauch et al., 1974] and atom interferometer

[Carnal and Mlynek, 1991; Keith et al., 1991]. However, in multiparticle

systems the superposition principle yields phenomena that are much richer

and more interesting than anything that can be seen in one-particle systems.

Quantum Entanglement, a simple name for superposition in a multipar-

ticle system, was first noticed by Schrodinger [Schrodinger, 1935] and since

then it has baffled generations of physicists. It is at the heart of the dis-

cussions of the Einstein-Podolsky-Rosen (EPR) paradox, of Bell’s inequality,

and of the non-locality of quantum mechanics [Einstein et al., 1935; Bell,

1964]. In recent years, entanglement has become a new focus of activity in

quantum physics because of immense theoretical and experimental progress

both in the foundation of quantum mechanics and in the new field of quantum

information science.

On the theoretical side, while the discovery of the conflict with local

realism following from Greenberger-Horne-Zeilinger (GHZ) entanglement of

three- or more particles [Greenberger et al., 1989; 1990] allows us to per-

5

6 CHAPTER 1. INTRODUCTION

form novel and completely new tests of local realism without inequalities,

the resource of entanglement has also many useful applications in quantum

information processing [Bennett, 1995], including quantum computation and

quantum communication. On the one hand, quantum computation, based on

a controlled manipulation of entangled states of quantum bits, might allow

us to build a new generation of computers which promise to be more pow-

erful than their classical counterparts [Deutsch, 1985], for example, Shor’s

discovery of quantum algorithms [Shor, 1994] enables us to factorize large

integers exponentially faster than the best known classical algorithms. On

the other hand, quantum communication schemes, such as quantum cryp-

tography [Bennett et al, 1992a], dense coding [Bennett and Wiesner, 1992]

and teleportation [Bennett et al., 1993], offer more efficient and secure ways

for the exchange of information in a network.

On the experimental side, the current technology is beginning to allow

us to manipulate rather than just observe individual quantum phenomena.

This opens up the possibility of realizing these above proposals in real experi-

ments. However, although there is fast progress in the theoretical description

of quantum information processing, the difficulties in handling quantum sys-

tems have not yet allowed an equal advance in the experimental realization of

the new proposals. Besides the promising developments of quantum cryptog-

raphy (the first provably secure way to send secret messages), peoples have

only recently succeeded in demonstrating the possibility of quantum dense

coding [Mattle et al., 1996], a way to quantum mechanically enhance data

compression. The main reason for this slow experimental progress is that,

although there exist methods to produce pairs of entangled photons [Kwiat

et al., 1995], entanglement has been demonstrated for atoms [Hagley et al.,

1997] only very recently and it has not been possible thus far to produce en-

tangled states of more than two quanta. Yet, all known methods of quantum

computation are applications of entanglement. The present experimental

challenge is therefore not to build a full-fledged universal quantum computer

straight away but rather to progress from experiments in which we merely

observe quantum interference and entanglement to experiments in which we

7

can control those quantum phenomena in the required way. All this above

leads to the main motivation of our experimental efforts in this dissertation.

Late in 1997, our group successfully achieved the first experimental demon-

stration of quantum teleportation [Bouwmeester, Pan et al., 1997] in which

we disembodied the polarization state of a photon into classical data and

EPR correlations, and then used these ingredients to reincarnate the state

in another photon which has never been anywhere near the first photon.

Due to our first realization of quantum teleportation, experimental research

in the field of quantum information is now attracting increasing attention

from both academia and industry. In the year of 1998, several groups in the

world achieved a series of important advances involving quantum computa-

tion and teleportation: In February, De Martini’s group [Boschi et al., 1998]

reported an optical realization for Popescu’s scheme–a variant of the original

teleportation proposal. In May, we [Pan et al., 1998b] experimentally real-

ized entanglement swapping, that is, teleportation of completely undefined

quantum state; Chuang and his coworkers, meanwhile, reported the first ex-

perimental realization of the Deutsch-Jozsa quantum algorithm using a bulk

nuclear magnetic resonance (NMR) technique [Chuang et al., 1998]. Then

in October, Kimble’s group [Furusawa et al., 1998] succeeded in teleporting

information on the amplitude and phase of an entire light beam to another

beam. In November, Nielsen et al. [Nielsen et al., 1998] teleported quan-

tum information from the nucleus of carbon atom to that of a neighboring

hydrogen atom.

Recently, according to the proposal for production of GHZ entanglement

out of entangled pairs [Zeilinger et al., 1997], we implemented a source of GHZ

entanglement for three spatially separated photons [Bouwmeester, Pan et al.,

1999], which is a further development of the technique that has been used

in our previous experiments for teleportation and entanglement swapping.

Such a source, for the first time, opens the door to demonstrate the GHZ

theorem. The first three-particle test of local realism without inequalities has

been done most recently [Pan et al., 1999a]. All these significant advances

8 CHAPTER 1. INTRODUCTION

greatly promote experimental research both in the foundations of quantum

mechanics and in the field of quantum information.

The aim of the thesis is to report the first experimental realization of

teleportation for arbitrary quantum states, to report the first observation of

GHZ entanglement for three spatially separated photons, and to report the

first demonstration of non-locality of quantum mechanics for nonstatistical

predictions of the theory. The main contents of the dissertation are organized

as follow:

Chapter 1 is a concise introduction to the applications of quantum entan-

glement. We briefly review the current theoretical and experimental advances

both in the foundation of quantum mechanics and in the new field of quantum

information science.

As all applications of entanglement necessitate both preparation and mea-

surement of entangled states, in Chapter 2, we shall theoretically describe

how quantum networks with simple quantum logic gates and a handful of

qubits allow us to control and manipulate quantum entanglement. Due to

the present absence of a general quantum gate, meanwhile we also present

a practical way to generate and identify entangled states, which constitutes

the basis of all experiments in the dissertation.

Quantum teleportation, the transmission and reconstruction over arbi-

trary distances of the state of a quantum system, is experimentally demon-

strated in Chapter 3. We describe in detail the theoretical and experimental

schemes of quantum teleportation. During teleportation, an initial photon

which carries the polarization that is to be transferred and one of a pair

of entangled photons are subjected to a measurement such that the second

photon of the entangled pair acquires the polarization of the initial pho-

ton. Quantum teleportation will be a critical ingredient for future quantum

computation networks.

Entanglement swapping enables one to entangle particles that never phys-

9

ically interacted with one another or which have never been dynamically cou-

pled by any other means. Chapter 4 reports in detail an optical experimental

realization of entanglement swapping. As we will see below, entanglement

swapping, besides its interest to fundamental physics, will have a number of

important applications in quantum communication.

In Chapter 5, we report the first observation of polarization entangle-

ment of three spatially separated photons. Such an entangled state is the

long-coveted GHZ state. In addition to facilitating more advanced forms of

quantum cryptography, our GHZ state will help provide a non-statistical test

of the foundations of quantum physics.

Chapter 6 is concerned with the test of local realism via a GHZ state.

Though previous experiments, based on observation of the entangled state of

two photons, have provided highly convincing evidence against local realism,

these ”Bell’s inequalities” tests require the measurement of many pairs of

entangled photons to build up a body of statistical evidence against the

idea. In contrast, the GHZ theorem described in the chapter shows that

in principle studying a single set of properties in the GHZ photons could

verify the predictions of quantum mechanics while contradicting those of

local realism. Using the source exploited in Chapter 5, we present here the

first experimental demonstration of the GHZ theorem.

Chapter 7 involves conclusions and outlooks. In the chapter, we briefly

summarize the results in the thesis, and discuss the differences among those

recent experimental advances of quantum teleportation. Finally, we also give

some prospects for future experiments.

Chapter 2

Manipulation of EntangledStates

It is well known that the preparation and measurement of Bell states is es-

sential for quantum dense coding [Bennett and Wiesner, 1992], for quantum

teleportation [Bennett et al., 1993], and for entanglement swapping [Zukowski

et al., 1993], the extension to the GHZ situation allows us to generalize the

two to the multi-particle case. This chapter consists of three sections. First,

we shall theoretically describe how using only single-quantum-bit (qubit)

operations and controlled-NOT gates [Barenco et al., 1995a; 1995b] one

can construct a suitable quantum network to produce and identify any of

the maximally entangled states for any number of particles [Bruss et al.,

1997]. Second, since until now such quantum networks have not yet been

built in the laboratory, we shall also describe the existing Bell-state analyzer

[Weinfurter, 1994; Braustein and Mann, 1995] and then present a universal

scheme and practically realizable procedures, by which one can readily iden-

tify two of the maximally entangled states of any number of photons [Pan

and Zeilinger, 1998]. At the end of this chapter a high intensity source of

polarization-entangled photon pairs [Kwiat et al., 1995], which has been used

in all experiments of the dissertation, is described, and some basic alignment

procedures are discussed briefly.

10

2.1. QUANTUM NETWORK AND ITS APPLICATIONS 11

Figure 2.1: Graphical representations of Hadamard and the quantumcontrolled-NOT gates. Here, a + b denotes addition modulo 2.

2.1 Quantum network and its applications

Generally, a bit is a classical system with two Boolean states 0 and 1, while

a qubit means a generic two-state quantum system with a chosen ”computa-

tional basis” {|0〉, |1〉} (e.g. the polarization of a photon or a spin-12particle).

A quantum logic gate is an elementary device which performs a fixed unitary

operation on selected qubits in a fixed period of time. Single-qubit quantum

logic gates are rather trivial and can be implemented, for example, by ex-

citing selected atomic transitions with laser pulses of controllable frequency,

intensity and duration. In fact, using a simple Hadamard gate and the quan-

tum controlled-NOT gates, one can prepare and identify any of N -particle

entangled states. The action of a Hadamard gate (Fig. 2.1a) is equivalent

to the following unitary transformation:

|0〉 → 1√2(|0〉+ |1〉)

|1〉 → 1√2(|0〉 − |1〉) (2.1)

12 CHAPTER 2. MANIPULATION OF ENTANGLED STATES

and the controlled-NOT gate(Fig. 2.1b) flips the second of two qubits if and

only if the first is |1〉, namely

|0〉|0〉 → |0〉|0〉|0〉|1〉 → |0〉|1〉|1〉|0〉 → |1〉|1〉|1〉|1〉 → |1〉|0〉

(2.2)

Consider now the network shown in Figure 2.2a. Under the action of the

gates on the left-hand side of the network, the input two-particle states will

undergo a series of unitary transformation. For example, if the input state

is |0〉|0〉, after passing through the two gates it will be transformed into:

|0〉|0〉Hadmard−−− −→ 1√

2(|0〉|0〉+ |1〉|0〉)

C−NOT 12−−− −→ 1√

2(|0〉|0〉|+ |1〉|1〉)

(2.3)

which is one of the four maximally entangled Bell states,

|Ψ±〉 = 1√2(|0〉 |1〉 ± |1〉 |0〉)

|Φ±〉 = 1√2(|0〉 |0〉 ± |1〉 |1〉) .

(2.4)

Correspondingly, the network could also prepare the two qubits in one of the

remaining three Bell states:

|1〉|0〉 −→ 1√2(|0〉|0〉| − |1〉|1〉) (2.5)

|0〉|1〉 −→ 1√2(|0〉|1〉|+ |1〉|0〉) (2.6)

|1〉|1〉 −→ 1√2(|0〉|1〉| − |1〉|0〉) (2.7)

It is easy to verify that the reversed quantum network (right-hand side of

Figure 2.2a) can be used to implement the so-called Bell measurement on the

2.1. QUANTUM NETWORK AND ITS APPLICATIONS 13

Figure 2.2: (a) The Bell measurement: the gates on the left hand side allowus to generate the four Bell states from the four possible different inputs.Reversing the order of the gates (right-hand side of the diagram) correspondsto a Bell measurement. (b) GHZ measurement: the same as in (a) for theeight GHZ states.


two qubits by disentangling the Bell states. In this way, the Bell measurement

is reduced to two single-particle measurements. The method can be directly

extended to a three-qubit case. Figure 2.2b shows how to prepare eight maxi-

mally entangled three-particle states, known as the GHZ states [Greenberger

1989,1990]. Reversing the procedure we obtain the unitary transformation

which reduces the GHZ measurement to the three single-particle measure-

ments.

We can write this in the following compact form, where a and b can each

take the values 0 and 1 and a and b denote NOT-a and NOT-b, respectively:

|0〉|a〉|b〉 ⇐⇒ 1√2

(|0〉|a〉||b〉+ |1〉|a〉|b〉

)(2.8)

|1〉|a〉|b〉 ⇐⇒ 1√2

(|0〉|a〉||b〉 − |1〉|a〉|b〉

)(2.9)

Let us also mention that the GHZ measurement provides an interesting

possibility of labeling the GHZ states via the corresponding binary output.

The three output bits then have the following meanings:

(i) The first output bit tells us whether the number of |0′〉 in the GHZ

state, written in the conjugate basis (this is given by |0′〉 = (1/√2)(|0〉+ |1〉)

and |1′〉 = (1/√2)(|0〉 − |1〉)), is even or odd. If the first output bit is |0〉,

there is an odd number of |0′〉s in the conjugate basis, otherwise an even

number.

(ii) The second output bit indicates whether the first two bits in the GHZ

superposition are the same or different. If the second output bit is |0〉, theyare the same.

(iii) The third output bit provides the same information with respect to

the first and third bit of the GHZ superposition.

Finally, we would like to emphasize that, in general, any measurement on

any number of qubits can be implemented using only single-qubit operations

2.2. PRACTICAL SCHEMES FOR ENTANGLED-STATE ANALYSIS 15

Figure 2.3: The beam splitter coherently transforms two input spatial modes(a, b) into two output spatial modes (c, d)

and the quantum controlled-NOT gates.

This follows from the fact that the quantum controlled-NOT gate, to-

gether with relatively trivial single-qubit operations, forms an adequate set

of quantum gates, i.e., the set from which any unitary operation may be

built [Barenco et al., 1995a; 1995b]. Thus if we want to measure observable

A pertaining to n qubits, we could construct a compensating unitary trans-

formation U which maps 2n states of the form |a1〉|a2〉...|an〉, where ai = 0, 1,

into the eigenstates of A. This allows both to prepare the eigenstates of A,

which in general can be highly entangled, and to reduce the measurement

described by A to n simple, single-qubit measurements.

2.2 Practical schemes for entangled-state anal-

ysis

Though quantum networks present a novel way to manipulate entangled

states, their experimental realization remains to be a challenge for the future

due to the lack of a quantum controlled-NOT gate. However, while no com-

plete Bell-state measurement procedure exists, we can already experimentally

identify two of the maximally entangled states of N photons.


2.2.1 Bell-state analysis

The Bell-state analyzer first suggested by Weinfurter et al. is based on the

two-photon interference effect at a 50:50 standard beam splitter. The beam

splitter has two spatial input modes a and b and two output modes c and

d (Fig. 2.3). Quantum mechanically, the action of the beam splitter on the

input modes can be written as

|a〉 −→ i√2|c〉+ 1√

2|d〉

|b〉 −→ 1√2|c〉+ i√

2|d〉 (2.10)

where, e.g. |a〉 describes the spatial quantum state of the particle in input

beam a. Eq. 2.10 describes the fact that the particle can be found with equal

probability (50%) in either of the output modes c and d, no matter through

which input beam it came. Here the factor i in Eq. 2.10 is a consequence of

unitarity. It corresponds physically to a phase jump upon reflection at the

semi-transparent mirror [Zeilinger, 1981]. Note that a standard beam splitter

is polarization independent, and thus has no effect on the polarization state

of the photon.

Let us now consider our beam splitter with two incident photons, 1 and

2, photon 1 in input beam a, and photon 2 in input beam b. Suppose that

photon 1 is in polarization state α|H〉1+β|V 〉1, and photon 2 is in polarizationstate γ|H〉2 + δ|V 〉2 ( here H and V denote horizontal and vertical linear

polarizations, and |α|2 + |β|2 = 1, |γ|2 + |δ|2 = 1). They each have the same

probability p = 0.5 to transmit the beam splitter or be reflected. Thus, four

different possibilities arise (Fig. 2.4).

(1) Both particles are reflected, (2) both particles are transmitted, (3) the

upper particle is reflected, the lower one is transmitted, and (4) the upper one

is transmitted and the lower one is reflected. Each of the four occurs with the

same probability, and one has to investigate now whether any interference

between these processes is possible. For distinguishable particles, for example

for classical ones, no interference arises and we thus arrive at the prediction


Figure 2.4: Two particles incident onto a beam splitter, one from each side.Four possibilities exist how the two particles can leave the beam splitter.


that in two of the cases, that is, with total probability p = 0.5, the two

particles end up in different output ports and, with probability p = 0.25, both

particles end up in the upper output beam and, with the same probability

p = 0.25, they end up in the lower output beam.

Let us now assume that the two photons have the same frequency and

arrive at the beam splitter simultaneously. As a result they are quantum

mechanically indistinguishable. In this case it is not possible, not even in

principle, to decide which of the incident particles ended up in a given output

port, we therefore have to consider coherent superpositions of the amplitudes

for these different possibilities. To show how the Bell-state analyzer works,

consider the input state

|ψi〉 = (α|H〉1 + β|V 〉1)|a〉1·(γ|H〉2 + δ|V 〉2)|b〉2.

(2.11)

where, for example, the first term in the equation indicates photon 1 with a

polarization state α|H〉1 + β|V 〉1 is in input mode a.

As shown in Eq. 2.10, for photons 1 and 2 passing through the beam split-

ter their spatial modes will undergo a corresponding unitary transformation.

The state in Eq. 2.11 thus evolves into

|ψf〉12 = 1√2(α|H〉1 + β|V 〉1)(i|c〉1 + |d〉1)·

1√2(γ|H〉2 + δ|V 〉2)(|c〉2 + i|d〉2).

(2.12)

It should be noted that photons 1 and 2 are not distinguishable anymore

after passing through the beam splitter. The total two-photon state including

both the spatial and the spin part, therefore, has to obey bosonic quantum

statistics. This implies that the outgoing physical state must be symmetric

under exchange of labels 1 and 2. To do so, one should symmetrize the state

|ψf〉12, that is, also include its exchange wave-function

|ψf〉21 = 1√2(α|H〉2 + β|V 〉2)(i|c〉2 + |d〉2)·

1√2(γ|H〉1 + δ|V 〉1)(|c〉1 + i|d〉1).

(2.13)


The final outgoing state therefore reads

|ψf〉 =1√2(|ψf〉12 + |ψf〉21), (2.14)

and consequently we have

|ψf〉 = 12√

2[(αγ + βδ)(|H〉1|H〉2 + |V 〉1|V 〉2) · i(|c〉1|c〉2 + |d〉1|d〉2)+(αγ − βδ)(|H〉1|H〉2 − |V 〉1|V 〉2) · i(|c〉1|c〉2 + |d〉1|d〉2)+(αδ + βγ)(|H〉1|V 〉2 + |V 〉1|H〉2) · i(|c〉1|c〉2 + |d〉1|d〉2)+(αδ − βγ)(|H〉1|V 〉2 − |V 〉1|H〉2) · (|d〉1|c〉2 − |c〉1|d〉2)].

(2.15)

As we will see below, Eq. 2.15 allows us to readily project the two-photon

state into two of the four maximally polarization-entangled states:

|Ψ±〉12 = 1√2(|H〉1 |V〉2 ± |V〉1 |H〉2)

|Φ±〉12 = 1√2(|H〉1 |H〉2 ± |V〉1 |V〉2) .

(2.16)

From Eq. 2.15, it is easy to verify that the two photons proceed after the

beam splitter in different emerging beams if, and only if, their polarization

state is in the state |Ψ−〉12 (refer to the fourth term of Eq. 2.15). Thus

we arrived at a possibility to identify one of the four Bell states, |Ψ−〉12,

uniquely on the basis that it is the only one which gives rise to a detection

of one photon in each of the outgoing beams of the beam splitter. For a

full analysis, we further need a way to distinguish between the other three

states, |Ψ+〉12, |Φ+〉12, and |Φ−〉12. Again, one can easily find that it is only in

the state |Ψ+〉12 that the two emerging photons have different polarizations.

In the two |Φ±〉12 states, they always share the same polarization. Thus,

a further step in Bell-state analysis implies that one inserts a two-channel

polarizer into each of the output ports of the beam splitter. Then, only the

state |Ψ+〉12 will give a coincidence count between the two output ports of

the polarizer on either side of the beam splitter. Yet, both |Φ±〉12 states will

give rise to the same joint detection of the two photons in either detector

after the final polarizer. By making use of two-particle interference effects

at a beam splitter, we can thus distinguish two of the four Bell states via

two-fold coincidence analysis.


2.2.2 GHZ-state analyzer

As an application of the concept of quantum erasure, we present in this part a

practical GHZ-state analyzer for identifying two of the N -particle entangled

states [Pan 1998a]. The basic elements of the experimental setup are just

polarizing beam splitters(PBS) and half-wave plates(HWP).

Before starting to discuss the GHZ-state analyzer, let’s first give a mod-

ified version of the Bell-state analyzer, which is similar to but different from

the former one shown in Fig. 2.3. Consider the arrangement of Fig. 2.5.

Two identical photons enter our Bell-state analyzer from modes A and B re-

spectively. Suppose they are in the most general polarization-superposition

state

|ψin〉 = α|HA〉|HB〉+ β|HA〉|VB〉+ γ|VA〉|HB〉+ δ|VA〉|VB〉 (2.17)

Because the polarizing beam splitter PBS transmits only the horizontal polar-

ization component and reflects the vertical component, after passing through

PBSAB the incident state will evolve into

|ψin〉 −→ α|HA2〉|HB1〉+ β|HA2〉|VB2〉+ γ|VA1〉|HB1〉+ δ|VA1〉|VB2〉 (2.18)

Where subscript ij(i = A,B, j = 1, 2) denotes the transformation from in-

put mode i to output mode j. Suppose that these two photons A and B

arrive at the polarizing beam splitter PBSAB simultaneously, and therefore

their spatial wavefuctions overlap each other. Then, according to the indis-

tinguishability of identical particles, we can directly denote HA2 as H2, VA1

as V1, HB1 as H1, and VB2 as V2. Thus, Eq. 2.18 reads

α|H1〉|H2〉+ β|H2〉|V2〉+ γ|V1〉|H1〉+ δ|V1〉|V2〉 (2.19)

Note that for the terms |H1〉|H2〉 and |V1〉|V2〉 the two photons are in

different output ports, and while for the terms |H2〉|V2〉 and |V1〉|H1〉 both


Figure 2.5: A modified Bell-state analyzer. Two indistinguishable photonsenter the Bell-state analyzer from input ports A and B. PBSAB, PBS1 andPBS2 are three polarizing beam splitters, which transmit the horizontal polar-ization component and reflect the vertical component. DH1, DV 1, DH2, andDV 2 are four photon-counting detectors.


photons are in the same output port. Therefore, we can identify states |ψd〉 =α|H1〉|H2〉+δ|V1〉|V2〉 and |ψs〉 = β|H2〉|V2〉+γ|H1〉|V1〉 using the coincidencebetween detectors in mode 1 and in mode 2. Obviously, in terms of the Bell

states, ψd can be rewritten as

|ψd〉 =1√2(α+ δ)|Φ+〉12 +

1√2(α− δ)|Φ−〉12 (2.20)

Thus, in order to finish the Bell-state measurement we now only need to

identify states |Φ+〉12 and |Φ−〉12, that is , we have to determine the relative

phase between terms of |H1〉|H2〉 and |V1〉|V2〉. Let the angle between the

HWP axis and the horizontal direction be 22.50 such that it corresponds to

a 450 rotation of the polarization. Therefore, for a photon passing the HWP,

its polarization state will undergo the following unitary transformation:

|Hi〉 −→ 1√2(|Hi〉+ |Vi〉)

|Vi〉 −→ 1√2(|Hi〉 − |Vi〉)

where i = 1, 2. Finally, |Φ+〉12 and |Φ−〉12 will thus be transformed into

|Φ+〉12 → |Φ+〉12 =1√2(|H1〉|H2〉+ |V1〉|V2〉) (2.21)

|Φ−〉12 → |Ψ+〉12 =1√2(|H1〉|V2〉+ |V1〉|H2〉) (2.22)

The above analysis show that we can easily identify two of the four incident

Bell states. Specifically, if we observe a coincidence either between detectors

DH1 and DH2 or DV 1 and DV 2, then the incident state was 1√2(|HA〉|HB〉 +

|VA〉|VB〉). On the other hand if we observe coincidence between detectors

DH1 and DV 2 or DV 1 and DH2, then the incident state was1√2(|HA〉|HB〉 −

|VA〉|VB〉). The other two incident Bell states will lead to no coincidence

between detectors in mode 1 and in mode 2. Such states are signified by

some kind of superposition of |HA〉|VB〉 and |VA〉|HB〉. This concludes our

demonstration that we can identify two of the four Bell states using the

coincidence between modes 1 and 2.


Figure 2.6: A GHZ-state analyzer. Three photons incident one each in modesA, B, and C will give rise to distinct 3-fold coincidence if they are in the GHZ-states |Φ+〉 or |Φ−〉. All the notations are the same as those in Fig. 2.5.

The reason why we discuss the modified version of Bell-state analyzer is

that the above scheme can directly be generalized to the N -particle case.

Making use of its basic idea, one can easily construct a type of GHZ-state

analyzer by which one can immediately identify two of the 2N maximally

entangled GHZ states.

For example, in the case of three identical photons, the eight maximally

entangled GHZ states are given by

|Φ±〉 = 1√2(|H〉|H〉|H〉 ± |V 〉|V 〉|V 〉) (2.23)

|Ψ1±〉 = 1√

2(|V 〉|H〉|H〉 ± |H〉|V 〉|V 〉) (2.24)


|Ψ2±〉 = 1√

2(|H〉|V 〉|H〉 ± |V 〉|H〉|V 〉) (2.25)

|Ψ3±〉 = 1√

2(|H〉|H〉|V 〉 ± |V 〉|V 〉|H〉) (2.26)

Where |Ψi±〉 designates that GHZ-state where the polarization of photon

i is different from the other two. Consider now the setup of Fig. 2.6 and

suppose that three photons enter the GHZ analyzer, each one from modes A,

B and C respectively. A suitable arrangement can be realized such that the

photon coming from mode A and the one coming from mode B overlap at

PBSAB and, thus they are correspondingly transformed into mode 1 and into

mode BC. Let’s further suppose that the photons from mode BC and mode

C overlap each other at PBSBC . Thus, following the above demonstration

for the case of the Bell-state analyzer, it is easy to find that the eight GHZ

states above will correspondingly evolve into

1√2(|H1〉|H2〉|H3〉 ± |V1〉|V2〉|V3〉) (2.27)

1√2(|H1〉|H3〉|V3〉 ± |V1〉|V2〉|H2〉) (2.28)

1√2(|H2〉|V2〉|H3〉 ± |H1〉|V1〉|V3〉) (2.29)

1√2(|H1〉|V1〉|H2〉 ± |V2〉|H3〉|V3〉) (2.30)

immediately after these three photons passed through PBSAB and PBSBC ,

and before they enter the half-wave plates(HWP). Here, e.g. Hi denotes a

photon with polarization H in output mode i.

From Eqs. 2.27-2.30, it is evident that one can observe three-fold coinci-

dence between modes 1, 2 and 3 only for the state of Eq. 2.27. For the other

states, there are always two particles in the same mode. We can thus dis-

tinguish the two states 1√2(|H〉|H〉|H〉± |V 〉|V 〉|V 〉) from the other six GHZ

states. Furthermore, after the states of Eq. 2.27 pass through the HWP, we

finally obtain


1√2(|H1〉|H2〉|H3〉+ |V1〉|V2〉|V3〉)→ 1

2(|H1〉|H2〉|H3〉+ |H1〉|V2〉|V3〉+|V1〉|H2〉|V3〉+ |V1〉|V2〉|H3〉)

(2.31)

1√2(|H1〉|H2〉|H3〉 − |V1〉|V2〉|V3〉)→ 1

2(|H1〉|H2〉|V3〉+ |H1〉|V2〉|H3〉+|V1〉|H2〉|H3〉+ |V1〉|V2〉|V3〉)

(2.32)

Thus, using three-fold coincidence we can readily identify the relative phase

between states |H〉|H〉|H〉 and |V 〉|V 〉|V 〉. This is because only the initial

state |Φ+〉 leads to coincidence between detectors DH1, DH2 and DH3 (or

H1V2V3, V1H2V3, V1V2H3). On the other hand, only the state |Φ−〉 leadsto coincidence between detectors DH1, DH2 and DV 3 (or H1V2H3, V1H2H3,

V1V2V3). Or, in conclusion, states |Φ+〉 and |Φ−〉 are identified by coinci-

dences between all three output modes 1, 2, and 3. They can be distinguished

because behind the half wave plates |Φ+〉 results in one or three horizontally,and zero or two vertically polarized photons, while |Φ−〉 results in zero or twohorizontally polarized photons and one or three vertically polarized ones.

Our GHZ-state analyzer has many possible applications. For example

the three photons entering via the modes A, B and C respectively could each

come from one entangled pair. Then projection of these three photons using

the GHZ-state analyzer onto the GHZ state Φ+ or Φ− implies that the other

three photons emerging from each pair will be prepared in a GHZ state.

It is clear that our scheme can readily be generalized to analyze entangled

states consisting of more than three photons by just adding more polarizing

beam splitters and half wave plates. Also, identification of analogs of our

scheme for GHZ-state analysis of atoms or of mode entangled states instead

of polarization entangled ones is straightforward.

Here, it is worth noting that an extension of the above scheme would pro-

vide us with a conditional GHZ-state source by which we can conveniently

observe four-particle GHZ correlations and further prepare three freely propa-

gating particles in a GHZ state [Zeilinger et al, 1997; Pan and Zeilinger, 1998;


Figure 2.7: A three- and four-photon polarization-entanglement source. Thephoton sources, A and B, pumped by short pulses, each one emits a photonpair in the superposition HH+V V . Then, one photon coming from source Aand one coming from source B overlap at the polarizing beam splitter PBS1.F is a narrow filter, PBS is a special polarizing beam splitter which transmits45◦ polarization and reflects −45◦ polarization. DT1 and DT2 are two single-photon detectors.


Pan and Zeilinger, 1999b]. Consider two sources A and B (see Fig. 2.7) each

one emitting a photon pair. Consider for simplicity that the photons emitted

by sources A and B are both in the same entangled state 1√2(|H〉|H〉+|V 〉|V 〉).

Then, using very analogous arguments as above we find that the state of the

four particles immediately after passage through the polarizing beam splitter

PBS1 will be the superposition

12(|H1〉|H2〉|H3〉|H4〉+|V1〉|V2〉|V3〉|V4〉+|H1〉|H3〉|V3〉|V4〉+|V1〉|V2〉|H2〉|H4〉)

(2.33)

Again, only for the superposition |H1〉|H2〉|H3〉|H4〉 + |V1〉|V2〉|V3〉|V4〉 wewould observe four-fold coincidence. Therefore, we then know these four par-

ticles are in the superposition |H1〉|H2〉|H3〉|H4〉+ |V1〉|V2〉|V3〉|V4〉 as soon aswe observe four-fold coincidence. Note that here the GHZ state is not directly

prepared but we know that the four particles are in a GHZ state under the

condition that one particle each is detected in each of the outgoing beams 1,

2, 3 and 4. This is a much weaker condition than any post-selection proce-

dure which might be based on properties of the particles. In an experiment

our case will not be distinguishable from the real situation occurring anyway

because of finite detector efficiency. That is, from a practical point of view,

even if one definitely prepares a full GHZ state one only will observe four-

fold coincidence in a fraction of time anyway. Thus, we conclude that using

our conditional GHZ-state one will be able to experimentally demonstrate

all features of a 4-particle GHZ state.

Yet, in the meantime we would like to note that our scheme in Fig. 2.7

also allows us to generate unconditional three-particle GHZ states via so-

called entangled entanglement [Krenn and Zeilinger, 1996]. For example,

one could analyze the polarization state of photon 2 by passing it through a

polarizing beam splitter PBS selecting 45◦ and −45◦) polarization. Then thepolarization state of the remaining three photons 1, 3 and 4 will be projected


into

1√2(|H1〉|H3〉|H4〉+ |V1〉|V3〉|V4〉

if and only if detector DT1 detects a single photon. Correspondingly, the

state of photons 1, 3 and 4 will be projected into

1√2(|H1〉|H3〉|H4〉 − |V1〉|V3〉|V4〉

if and only if detector DT2 detects a single photon. In the scheme, the

detection of photon 2 actually plays the double role of both getting rid of the

last two terms in Eq. 2.33 and projecting the remaining three photons into

a spatially separated and freely propagating GHZ state. Such a GHZ-state

could be extremely useful both in further test of local realism versus quantum

mechanics and in future application of third-man cryptography.

Finally we would like to note that in all these schemes we have used the

principle of quantum erasure in a way that behind our GHZ-state analyzer at

least some of the photons registered cannot be identified anymore as to which

source they came from. This implies very specific experimental schemes, be-

cause the particles might have been created at different times. One theo-

retical possibility is to apply the principle of ultra-coincidence [Zukowski et

al., 1993]. This means that the photons must be registered within a time

short compared to their coherence time. For practical reasons, i.e. the un-

availability of sufficiently fast detectors, the scheme cannot be realized at

present. Yet, alternatively, one can create the particles within a time inter-

val small compared to their coherence times. This in practice implies the use

of pulsed sources and of filters behind them which introduce coherence times

larger than the pulse length [Zukowski et al., 1995]. Such a scheme has been

successfully used in our following experiments on quantum teleportation, en-

tanglement swapping and three-particle GHZ entanglement.

2.3. POLARIZATION-ENTANGLED PHOTON PAIRS 29

2.3 Polarization-entangled photon pairs

Because of the same reason, i.e. the absence of a quantum logic gate, at the

moment there are not so many possibilities to create states of entangled par-

ticles in the laboratory. Fortunately, the process of spontaneous parametric

down-conversion provides mechanisms by which pairs of entangled photons

can be produced with reasonable intensity and in good purity. In the down-

conversion process, one uses a non-centrolsymmetric crystal with nonlinear

electric susceptibility. In such a medium, an incoming photon can decay

with relatively small probability into two photons in a way that energy and

momentum inside the crystal are conserved.

Here we will describe a simple technique to produce polarization-entangled

photon pairs using the process of noncollinear type-II parametric down-

conversion [Kwiat et al., 1995]. In the experiment, the desired polarization-

entangled state is produced directly out of a single nonlinear crystal [BBO

(beta-barium-borate)]. In that process, the two photons are emitted with dif-

ferent polarizations(Fig. 2.8). Calculating the emission direction of the pho-

tons [Caro and Garuccio, 1994; Kwiat et al, 1995], one notices that photons

of each polarization are emitted into one cone in such a way that momenta of

two photons always add up to the momentum of the pump photon. Thus, the

emission direction of each individual photon is completely uncertain within

the cone, but once one photon is registered, and thus its emission direction

is defined, the other photon is found just exactly opposite from the pump

beam on the other cone. The total quantum mechanical state is therefore

extremely rich and is a superposition of all such pairs of emission modes.

The interesting point is now that the crystal can be cut and arranged

such that the two cones intersect, as shown in Fig. 2.8. Then, along the lines

of intersection, the polarization of neither photon is defined, but what is de-

fined is the fact that the two photons have to have different polarizations.

This contains all the necessary features of entanglement in a nutshell. Mea-

surement on each of the photons separately is totally random and gives with


Figure 2.8: Principle of type-II parametric down-conversion. Inside a nonlin-ear crystal(here, BBO), an incoming pump photon can decay spontaneouslyinto two photons. Two down-converted photons arise polarized orthogonallyto each other. Each photon is emitted into a cone, and the photon on the topcone is vertically polarized while its exactly opposite partner in the bottomcone is horizontally polarized. Along the directions where the two cones in-tersect, their polarizations are undefined; all that is known is that they haveto be different, which results in polarization entanglement between the twophotons in beams A and B.

2.3. POLARIZATION-ENTANGLED PHOTON PAIRS 31

equal probability vertical or horizontal polarization. But once one photon,

for example photon A, is measured, the polarization of the other photon B

is orthogonal! Choosing an appropriate basis, e.g. |H〉 and |V 〉, the stateemerging through the two beams A and B thus is a superposition of |H〉|V 〉and |V 〉|H〉, say

1√2(|H〉A|V 〉B + eiα|V 〉A|H〉B) (2.34)

where the relative phase α arises from the crystal birefringence, and an overall

phase shift is omitted.

Using an additional birefringent phase shifter (or even slightly rotating

the down-conversion crystal itself), the value of α can be set as desired, e.g.,

to the values 0 or π. Somewhat surprisingly, a net phase shift of π may

be obtained by a 90◦ rotation of a quarter wave plate in one of the paths.

Similarly, a half wave plate in one path can be used to change horizontal

polarization to vertical and vice versa. One can thus very easily produce any

of the four EPR-Bell states in Eq. 2.16.

The birefringent nature of the down-conversion crystal complicates the

actual entangled state produced, since the ordinary and extraordinary pho-

tons have different velocities inside the crystal, and propagate along different

directions even though they become collinear outside the crystal (an effect

well known from calcite prisms, for example). The resulting longitudinal and

transverse walk-offs between the two terms in the state (2.34) are maximal

for pairs created near the entrance face, which consequently acquire a relative

time delay δT = L(1/uo − 1/ue) (L is the crystal length, and uo and ue are

the ordinary and extraordinary group velocities, respectively) and a relative

lateral displacement d = L tan ρ (ρ is the angle between the ordinary and

extraordinary beams inside the crystal). If δT ≥ τc, the coherence time of

the down-conversion light, then the terms in Eq. 2.34 become, in principle,

distinguishable by the order in which the detectors would fire, and no inter-

ference will be observable. Similarly, if d is larger than the coherence width,


the terms can become partially labeled by their spatial location.

Because the photons are produced coherently along the entire length of

the crystal, one can completely compensate for the longitudinal walk-off [Ru-

bin et al., 1994]—after compensation, interference occurs pairwise between

processes where the photon pair is created at distances ±x from the middle

of the crystal. The ideal compensation is therefore to use two crystals, one in

each path, which are identical to the down-conversion crystal, but only half

as long. If the polarization of the light is first rotated by 90◦ (e.g., with a

half wave plate), the retardations of the o and e components are exchanged

and complete temporal indistinguishability is restored (δT = 0). The same

method provides optimal compensation for the transverse walk-off effect as

well. Here, the compensation crystals were oriented along the same direc-

tion as that of the down-conversion crystal. In the following experiments we

always slightly rotate the orientation of one of the compensation crystals to

tune the relative phase α = π.

The BBO crystal used in our experiments is 2.0mm long and was cut at

θpm = 43.5◦ (the angle between the crystal optic axis and the pump) in order

to result in a well-defined the intersection between the two cones. The two

cone-overlap directions, selected by irises before the detectors, were conse-

quently separated by 5◦ when the pump beam is precisely orthogonal to the

surface of the crystal. The transverse walk-off d (0.2mm) was small com-

pared to the coherent pump beam width (2mm), so the associated labeling

effect was minimal. However, it was necessary to compensate for longitudinal

walk-off, since our 2.0mm BBO crystal produced δT = 260fs, while τc [de-

termined by the collection irises and interference filters (centered at 788nm,

4.6nm FWHM)] was about of the same order. As discussed above, we used

an additional BBO crystal (1.0mm thickness, θpm = 43.5◦) in each of the

paths, preceded by a half wave plate to exchange the roles of the horizontal

and vertical polarizations.

Chapter 3

Quantum Teleportation

3.1 Introduction

The dream of teleportation is to be able to travel by simply reappearing at

some distant location. An object to be teleported can be fully characterized

by its properties, which in classical physics can be determined by measure-

ment. To make a copy of that object at a distant location one does not

need the original parts and pieces–all that is needed is to send the scanned

information so that it can be used for reconstructing the object. But how

precisely can this be a true copy of the original? What if these parts and

pieces are electrons, atoms and molecules? What happens to their individual

quantum properties, which according to Heisenberg’s uncertainty principle

cannot be measured with arbitrary precision?

Bennett et al. [Bennett et al., 1993] have suggested that it is possible to

transfer the quantum state of a particle onto another particle– the process of

quantum teleportation–provided one does not get any information about the

state in the course of this transformation. This requirement can be fulfilled

by using entanglement, according to Schrodinger, the essential feature of

quantum mechanics [Schrodinger, 1935]. It describes correlations between

quantum systems much stronger than any classical correlation could be.

33

34 CHAPTER 3. QUANTUM TELEPORTATION

The possibility of transferring quantum information is one of the cor-

nerstones of the emerging field of quantum communication and quantum

computation [Bennett, 1995]. As we will see below, quantum teleportation

is indeed not only a critical ingredient for quantum computation and com-

munication, its experimental realization will also allow new studies of the

fundamentals of quantum theory.

In the present chapter, we report the first experimental verification of

quantum teleportation [Bouwmeester, Pan et al., 1997]. By producing pairs

of entangled photons by the process of parametric down-conversion and using

two-photon interferometry for analyzing entanglement, one could transfer

a quantum property (in our case the polarization state) from one photon

to another. The methods developed for this experiment will be of great

importance both for exploring the field of quantum information as well as

for future experiments on the foundations of quantum mechanics.

3.2 Quantum teleportation–the idea

3.2.1 The problem

To make the problem of transferring quantum information clearer suppose

that Alice has some particle in a certain quantum state |Ψ〉 and she wants

Bob, at a distant location, to have a particle in that state. There is certainly

the possibility to send Bob the particle directly. But suppose that the com-

munication channel between Alice and Bob is not good enough at the time

of the procedure to preserve the necessary quantum coherence or suppose

that this would take too much time, which could easily be the case if |Ψ〉 isthe state of a more complicated or massive object. Then, what strategy can

Alice and Bob pursue?

As mentioned above, no measurement that Alice can perform on |Ψ〉 willbe sufficient for Bob to reconstruct the state because the state of a quantum

3.2. QUANTUM TELEPORTATION–THE IDEA 35

system cannot be fully determined by measurements. Quantum systems are

so evasive because they can be in a superposition of several states at the same

time. A measurement on the quantum system will force it into only one of

these states; this is often referred to as the projection postulate. We can

illustrate this important quantum feature by taking a single photon, which

can be horizontally or vertically polarized, indicated by the states |H〉 and|V 〉. It can even be polarized in the general superposition of these two states

|Ψ〉 = α |H〉+ β |V 〉 , (3.1)

were α and β are two complex numbers satisfying |α|2 + |β|2 = 1. To place

this example in a more general setting we can replace the states |H〉 and|V 〉 in Eq.(3.1) by |0〉 and |1〉, which refer to the states of any two-state

quantum system. Superpositions of |0〉 and |1〉 are called qubits to signify

the new possibilities introduced by quantum physics into information science

[Schumacher, 1995].

If a photon in state |Ψ〉 passes through a polarizing beamsplitter, a devicethat reflects (transmits) horizontally (vertically) polarized photons, it will be

found in the reflected (transmitted) beam with probability |α|2 (|β|2). Thenthe general state |Ψ〉 has been projected either onto |H〉 or onto |V 〉 bythe action of the measurement. We conclude that the rules of quantum

mechanics, in particular the projection postulate, make it impossible for

Alice to perform a measurement on |Ψ〉 by which she would obtain all the

information necessary to reconstruct the state.

3.2.2 The concept of quantum teleportation

Although the projection postulate in quantum mechanics seems to bring

Alice’s attempts to provide Bob with the state |Ψ〉 to a halt, it was realizedby Bennett et al. [Bennett et al., 1993] that precisely this projection postulate

enables teleportation of |Ψ〉 from Alice to Bob. During teleportation Alice


will destroy the quantum state at hand while Bob receives the quantum state,

with neither Alice nor Bob obtaining information about the state |Ψ〉. A key

role in the teleportation scheme is played by an entangled ancillary pair of

particles which will be initially shared by Alice and Bob.

� � �

classicalinformation

ALICE

BOB

EPR-source

teleportedstate

entangled pair

initialstate

BSM

U

Figure 3.1: Scheme showing principle of quantum teleportation. Alice has aquantum system, particle 1, in an initial state which she wants to teleport toBob. Alice and Bob also share an ancillary entangled pair of particles 2 and3 emitted by an Einstein-Podolsky-Rosen(EPR) source. Alice then performsa joint Bell-state measurement (BSM) on the initial particle and one of theancillaries, projecting them also onto an entangled state. After she has sentthe result of her measurement as classical information to Bob, he can performa unitary transformation (U) on the other ancillary particle resulting in itbeing in the state of the original particle.

Suppose particle 1 which Alice wants to teleport is in the initial state

|Ψ〉1 = α |H〉1 + β |V 〉1 (Fig. 3.1), and the entangled pair of particles 2 and

3 shared by Alice and Bob is in the state:

∣∣∣Ψ−⟩23=

1√2(|H〉2 |V 〉3 − |V 〉2 |H〉3) . (3.2)


That entangled pair is a single quantum system in an equal superposition of

the states |H〉2 |V 〉3 and |V 〉2 |H〉3. The entangled state contains no informa-tion on the individual particles; it only indicates that the two particles will

be in opposite states. The important property of an entangled pair is that as

soon as a measurement on one of the particles projects it, say, onto |H〉 thestate of the other one is determined to be |V 〉, and vice versa. How could

a measurement on one of the particles instantaneously influence the state

of the other particle which, can be arbitrarily far away?! Einstein, among

many other distinguished physicists, could simply not accept this ”Spooky

action at a distance”. But this property of entangled states has now been

demonstrated by numerous experiments (for reviews, see refs. [Clauser and

Shimony, 1978; Greenberger et al., 1993].

The teleportation scheme works as follows. Alice has the particle 1 in the

initial state |Ψ〉1 and the ancillary particle 2. Particle 2 is entangled with theother ancillary particle 3 in the hands of Bob. Although this establishes the

possibility of nonclassical correlations between Alice and Bob, the entangled

pair at this stage contains no information about |Ψ〉1. Indeed the entire

system, comprising Alice’s unknown particle 1 and the entangled pair is in

a pure product state, |Ψ〉1 |Ψ−〉23, involving neither classical correlation nor

quantum entanglement between the unknown particle and the entangled pair.

Therefore no measurement on either member of the entangled pair, or both

together, can yield any information about |Ψ〉1.

The essential point to achieve teleportation is to perform a joint Bell-state

measurement on particles 1 and 2 which projects them onto one of the four

entangled states :

|Ψ±〉12 = 1√2(|H〉1 |V〉2 ± |V〉1 |H〉2)

|Φ±〉12 = 1√2(|H〉1 |H〉2 ± |V〉1 |V〉2) .

(3.3)

Note that these four states are a complete orthonormal basis for particles 1

and 2. The complete state of the three particles before Alice’s measurement


is

|Ψ〉123 = α√2(|H〉1 |H〉2 |V〉3 − |H〉1 |V〉2 |H〉3)

+ β√2(|V〉1 |H〉2 |V〉3 − |V〉1 |V〉2 |H〉3) .

(3.4)

In this equation, each direct product |〉1 |〉2 can be expressed in terms of thefour Bell states, and one can thus rewrite Eq.(3.4) as

|Ψ〉123 = 12[|Ψ−〉12 (−α |H〉3 − β |V〉3)

+ |Ψ+〉12 (−α |H〉3 + β |V〉3)+ |Φ−〉12 (α |V〉3 + β |H〉3)+ |Φ+〉12 (α |V〉3 − β |H〉3)] .

(3.5)

It follows that, regardless of the unknown state |Ψ〉1, the four Bell-state

measurement outcomes are equally likely, each occurring with probability

1/4. Quantum physics predicts that once particles 1 and 2 are projected into

one of the four entangled states, particle 3 is instantaneously projected into

one of the four pure states superposed in Eq.(3.5). Denoting |H〉 by the

vector

(10

)and |V 〉 by

(01

), they are thus, respectively,

− |Ψ〉3 ,(−1 00 1

)|Ψ〉3 ,(

0 11 0

)|Ψ〉3 ,

(0 −11 0

)|Ψ〉3 .

(3.6)

where |Ψ〉3 = α |H〉3+β |V 〉3. Each of these possible resultant states for Bob’sEPR particle 3 is related in a simple way to the original state |Ψ〉1 which

Alice sought to teleport. In the case of the first (singlet) outcome, the state of

particle 3 is the same as the initial state of particle 1 except for an irrelevant

phase factor, so Bob need do nothing further to produce a replica of Alice’s

unknown state. In the other three cases, Bob could accordingly apply one of

the unitary transformations in Eq.(3.6) to convert the state of particle 3 into

the original state of particle 1, after receiving via a classical communication

channel the information which of the Bell-state measurement results was


obtained by Alice. (For the photon polarization state, one can use a suitable

combination of half-wave plates to perform these unitary transformations.)

After Bob’s unitary operation, the final state of particle 3 is therefore

|Ψ〉3 = α |H〉3 + β |V 〉3 . (3.7)

Note that during the Bell-state measurement particle 1 loses its identity

because it becomes entangled with particle 2. Therefore the state |Ψ〉1 is

destroyed on Alice’s side during teleportation.

The result in Eq.(6.2) deserves some further comments. The transfer of

quantum information from particle 1 to particle 3 can happen over arbitrary

distances, hence the name teleportation. Experimentally, quantum entangle-

ment has been shown to survive over distances of the order of 10 km [Tittel

et al., 1998a; 1998b]. We note that in the teleportation scheme it is not

necessary for Alice to know where Bob is. Furthermore, the initial state of

particle 1 can be completely unknown not only to Alice but to anyone. It

could even be quantum mechanically completely undefined at the time the

Bell-state measurement takes place. This is the case when, as already re-

marked by Bennett et al. [Bennett et al., 1993], particle 1 itself is a member

of an entangled pair and therefore has no well-defined properties on its own.

This ultimately leads to entanglement swapping [Zukowski et al., 1993, Bose

et al., 1998].

It is also important to notice that the Bell-state measurement does not

reveal any information on the properties of any of the particles. This is the

very reason why quantum teleportation using coherent two-particle superpo-

sitions works, while any measurement on one-particle superpositions would

fail. The fact that no information whatsoever on either particle is gained

is also the reason why quantum teleportation escapes the verdict of the no-

cloning theorem [Wootters and Zurek, 1982]. After successful teleportation

particle 1 is not available in its original state anymore, and therefore particle

3 is not a clone but really the result of teleportation.


3.3 Experimental teleportation

3.3.1 Experimental scheme

Teleportation necessitates both production and measurement of entangled

states; these are the two most challenging tasks for any experimental realiza-

tion. Thus far there are only a few experimental techniques by which one can

prepare entangled states, and there exists no experimentally realized proce-

dure to identify all four Bell-states for any kind of quantum system composed

of two separate particles. However, as discussed in Chapter 2 entangled pairs

of photons can readily be generated and they can be projected onto at least

two of the four Bell states.

Using the technique in section 2.3, we were able to produce the entangled

photons 2 and 3 by type II parametric down-conversion (Fig. 3.2). Inside a

non-linear crystal, an incoming pump photon can decay spontaneously into

two photons which are in the polarization entangled state given by equation

(3.2).

For practical convenience, in the experiment we decided to analyze only

the projection onto |Ψ−〉12 . As discussed in section 2.2.1, this projection is

realized by interfering the two photons, 1 and 2, at a 50 : 50 beam splitter

and, detecting a coincidence between the two detectors at the different out-

put ports of the beam splitter. Here, such a coincident detection acts as a

projection onto |Ψ−〉12. Since originally the polarization state of photon 2 is

completely undetermined, the combined state between photons 1 and 2 is in

an equal superposition of the four Bell states. As a result, in one out of four

cases on average a coincidence will be recorded by the two detectors behind

the beam splitter; that is, a projection onto |Ψ−〉12 takes place.

It is clear that once particles 1 and 2 are projected into |Ψ−〉12, particle

3 is instantaneously projected into |Ψ〉3. Yet we note, with emphasis, that

even we choose to identify only one of the four Bell states, here |Ψ−〉12,

3.3. EXPERIMENTAL TELEPORTATION 41

Figure 3.2: Experimental Setup. A pulse of ultraviolet light passing througha non-linear crystal creates the ancillary pair of entangled photons 2 and 3.After retroflection during its second passage through the crystal the ultravioletpulse creates another pair of photons, one of which will be prepared in theinitial state of photon 1 to be teleported, the other one serves as a triggerindicating that a photon to be teleported is under way. Alice then looks forcoincidences after a beam splitter (BS) where the initial photon and one ofthe ancillaries are superposed. Bob, after receiving the classical informationthat Alice obtained a coincidence count in detectors f1 and f2 identifying the|Ψ−〉12 Bell-state, knows that his photon 3 is in the initial state of photon 1which he then can check using polarization analysis with the polarizing beamsplitter (PBS) and the detectors d1 and d2. The detector P provides theinformation that photon 1 is under way.


teleportation is successfully achieved, albeit only in a quarter of the cases.

As mentioned already, if we further insert a two-channel polarizer into each of

the outputs of the beam splitter BS, then a coincident detection between the

two outputs of the polarizer on either side of the BS acts as a projection onto

|Ψ+〉12. Thus, a slight change of our scheme is actually capable of achieving

teleportation in 50% of the time–in those occasions when Alice happened to

detect state |Ψ−〉12 or |Ψ+〉12.

Note that in our teleportation scheme the Bell-state analysis relies on

the interference of two independently created photons. One, therefore, has

to guarantee good spatial and temporal overlap at the beam splitter and,

above all, one has to erase all kinds of path information for photons 1 and 2.

Especially the strong time and frequency correlations of the two photons 2

and 3 created by parametric down-conversion can give rise to Welcher-Weg

information for the interfering photons [Herzog et al., 1995].

There are two possibilities for quantum erasure. In the first one, Welcher-

Weg information is erased by detecting photons 1 and 2 within time intervals

much shorter than their coherence time [Zukowski et al., 1993]. Then, such

ultracoincident registrations are too close in time to discriminate which of

the detected photons shares the source with photon 3 or with photon 4,

respectively. Yet, this method cannot be used in practice due to the poor

time resolution of the existing single-photon detectors (typically 0.5ns for

Si-avalanche photodiodes as compared to typical coherence times of about

500fs).

The second possibility involves increasing the coherence times of the in-

terfering photons to become much longer than the time interval within which

they are created [Zukowski et al., 1995]. Then again, one cannot infer any-

more which of the detected photons 1 or 2 was created together with photon

3, or with photon 4, respectively. In our experiment UV pulses with a du-

ration of 200fs are used to create the photon pairs. We then chose narrow

bandwidth filters (∆λ = 4.6nm) in front of the detectors registering photons

1 and 2. The resulting coherence time of about 500fs is sufficiently longer


than the pump pulse duration. Furthermore, single-mode fiber couplers act-

ing as spatial filters were used to guarantee good mode overlap of the detected

photons.

Figure 3.2 is a schematic drawing of the experimental setup. UV pulses

are produced by frequency doubling the pulses of a commercial mode locked

Ti:sapphire laser from 788 to 394nm using a nonlinear LBO crystal (LiB3O5).

For a repetition rate of 76 MHz we obtained an averaged power of 500 mW.

Passing the UV pulses through a BBO crystal (β−BaB2O4) creates via type-

II down-conversion a pair of photons, 2 and 3, in the entangled state |Ψ−〉23.

After reflection, the pump pulse passes the crystal again and produces the

second pair of photons 1 and 4. Photon 1 is prepared in the initial state to be

teleported, and its partner, photon 4, serves to indicate that it was emitted.

How can one experimentally prove that an arbitrary unknown quantum

state can be teleported? First, one has to show that teleportation works for

a (complete) basis, a set of known states into which any other state can be

decomposed. A basis for polarization states has just two components, and

in principle we could choose as the basis horizontal and vertical polarization

as emitted by the source. Yet this would not demonstrate that teleporta-

tion works for any general superposition, because these two directions are

preferred directions in our experiment. Therefore, in the first demonstration

we chose as the basis for teleportation the two states linearly polarised at

-45◦ and +45◦ which are already superpositions of the horizontal and vertical

polarizations. Second, one has to show that teleportation works for superpo-

sitions of these base states. Therefore we also demonstrate teleportation for

circular polarisation. This covers all three mutually orthogonal axes of the

Poincare sphere.

3.3.2 Results

In the first experiment photon 1 is polarized at 45◦. Teleportation should

work as soon as photon 1 and 2 are detected in the |Ψ−〉12 state, which occurs


in 25% of all possible cases. The |Ψ−〉12 state is identified by recording a

coincidence between two detectors, f1 and f2, placed behind the beamsplitter

(Fig. 3.2).

If we detect a f1f2 coincidence (between detectors f1 and f2), then photon

3 should also be polarized at 45◦. The polarization of photon 3 is analysed by

passing it through a polarizing beamsplitter selecting +45◦ and −45◦ polar-ization. To demonstrate teleportation, only detector d2 at the +45◦ output

of the polarizing beamsplitter should click (that is, register a detection) once

detectors f1 and f2 click. Detector d1 at the −45◦ output of the polarizingbeamsplitter should not detect a photon. Therefore, recording a three-fold

coincidence d2f1f2 (+45◦ analysis) together with the absence of a three-fold

coincidence d1f1f2 (−45◦ analysis) is a proof that the polarization of photon1 has been teleported to photon 3.

To meet the condition of temporal overlap, we change in small steps the

arrival time of photon 2 by changing the delay between the first and second

down conversion by translating the retroflection mirror (Fig. 3.2). In this

way we scan into the region of temporal overlap at the beamsplitter so that

teleportation should occur.

Outside the region of teleportation photon 1 and 2 each will go either to f1

or to f2 independent of one another. The probability to have a coincidence

between f1 and f2 is therefore 50%, which is twice as high as inside the

region of teleportation. Photon 3 should not have a well-defined polarization

because it is part of an entangled pair. Therefore, d1 and d2 have both

a 50% chance of receiving photon 3. This simple argument yields a 25%

probability both for the −45◦ analysis (d1f1f2 coincidences) and for the +45◦

analysis (d2f1f2 coincidences) outside the region of teleportation. Figure 3

summarizes the predictions as function of the delay. Successful teleportation

of the +45◦ polarization state is then characterized by a decrease to zero in

the −45◦ analysis (Fig. 3.3a), and by a constant value for the +45◦ analysis(Fig. 3.3b).


-100 -50 0 50 100

(b)+45°

Theory: +45° Teleportation3

-fo

ldc

oin

cid

en

ce

pro

ba

bili

ty

Delay (µm)

0,00

0,05

0,10

0,15

0,20

0,25

0,00

0,05

0,10

0,15

0,20

0,25

(a)-45°

Figure 3.3: Theoretical prediction for the three-fold coincidence probabilitybetween the two Bell-state detectors (f1, f2) and one of the detectors analysingthe teleported state. The signature of teleportation of a photon polarizationstate at +45◦ is a dip to zero at zero delay in the three-fold coincidence ratewith the detector analysing -45◦ (d1f1f2) (a) and a constant value for thedetector analysis +45◦ (d2f1f2) (b). The shaded area indicates the region ofteleportation.


The theoretical prediction of Fig. 3.3 may easily be understood by real-

izing that at zero delay there is a decrease to half in the coincidence rate

for the two detectors of the Bell-state analyser, f1 and f2, as compared to

outside the region of teleportation. Therefore, if the polarization of photon 3

were completely uncorrelated to the others the three-fold coincidence should

also show this dip to half. That the right state is teleported is indicated by

the fact that the dip goes to zero in Fig. 3.3a and it is filled to a flat curve

in Fig. 3.3b.

We note that about as likely as production of photons 1, 2, and 3 is emis-

sion of two pairs of down-converted photons by a single source. Although

there is no photon coming from the second source (photon 1 is absent), there

will still be a significant contribution to the three-fold coincidence rates.

These coincidences have nothing to do with teleportation and can be identi-

fied by blocking the path of photon 1.

The probability for this process to yield spurious two- and three-fold co-

incidences can be estimated by taking into account the experimental param-

eters. The experimentally determined value for the percentage of spurious

three-fold coincidences is 68%± 1%. In the experimental graphs of Fig. 3.4

we have subtracted the experimentally determined spurious coincidences.

The experimental results for teleportation of photons polarized under

+45◦ is shown in the left column of Fig. 3.4; Fig. 3.4a and b should be

compared with the theoretical predictions as shown in Fig. 3.3. The strong

decrease in the −45◦ analysis, and the constant signal for the +45◦ analysis,indicate that photon 3 is polarized along the direction of photon 1, confirming

teleportation.

The results for photon 1 polarized at −45◦ demonstrate that teleporta-tion works for a complete basis for polarization states (right-hand column

of Fig. 3.4). To rule out any classical explanation for the experimental re-

sults, we have produced further confirmation that our procedure works by

additional experiments. In these experiments we teleported photons linearly


-fo

ldc

oin

cid

en

ce

sp

er

20

00

sec

on

ds

(c)

-45° Teleportation+45° Teleportation

0

100

200

300

400

500

600

0

100

200

300

400

500

(a)

(b)

-150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150

0

100

200

300

400

500

600

0

100

200

300

400

500

(d)

Delay (µm) Delay (µm)

-45°

+45°

-45°

+45°

Figure 3.4: Experimental results. Measured three-fold coincidence ratesd1f1f2 (-45◦) and d2f1f2 (+45◦) in the case that the photon state to be tele-ported is polarized at +45◦ (a and b) or at -45◦ (c and d). The coincidencerates are plotted as function of the delay (in µm) between the arrival of pho-ton 1 and 2 at Alice’s beamsplitter (see Fig.1.2). The three-fold coincidencerates are plotted after subtracting the spurious three-fold background contri-bution (see text). These data, compared with Fig.1.3, together with similarones for other polarisations (Table 1) confirm teleportation for an arbitrarystate.


polarized at 0◦ and at 90◦, and also teleported circularly polarized photons.

The experimental results are summarized in Table 1, where we list the visibil-

ity of the dip in three-fold coincidences, which occurs for analysis orthogonal

to the input polarization.

polarization visibility+45◦ 0.63± 0.02−45◦ 0.64± 0.020◦ 0.66± 0.0290◦ 0.61± 0.02

circular 0.57± 0.02

Table 1

As mentioned above, the values for the visibilities are obtained after sub-

tracting the offset caused by spurious three-fold coincidences. These can

experimentally be excluded by conditioning the three-fold coincidences on

the detection of photon 4, which effectively projects photon 1 into a single-

particle state. We have performed this four-fold coincidence measurement for

the case of teleportation of the +45◦ and +90◦ polarization states, that is, for

two non-orthogonal states. The experimental results are shown in Fig. 3.5.

Visibilities of 70%± 3% are obtained for the dips in the orthogonal polariza-

tion states! Here, these visibilities are directly the degree of polarization of

the teleported photon in the right state without any background subtracted.

As can be seen from the measured visibilities, the teleportation fidelity is

rather high in our experiment. Typically, it is of the order of 0.85. This very

clearly surpasses the limit of 2/3 [Massar and Popescu, 1995] which at best

could have been obtained by Alice performing a polarisation measurement on

the given photon, informing Bob about the measurement result via classical

communication, and by Bob repreparing the photon state at his output.

The measured high fidelity proves that we demonstrated teleportation of the

quantum state of a single photon.


(c)(a)

(b) (d)

-45° 0°

+45° +90°

-150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150

4-f

old

co

inc

ide

nc

es

pe

r4

00

0se

co

nd

s

90° Teleportation45° Teleportation

Delay (µm) Delay (µm)

0

20

40

60

80

100

0

20

40

60

80

100

0

20

40

60

80

100

120

0

20

40

60

80

100

120

Figure 3.5: Four-fold coincidence rates (without background subtraction).Conditioning the three-fold coincidences as shown in Fig.1.4 on the registra-tion of photon 4 (see Fig.1.2) eliminates the spurious three-fold background.Graphs a and b show the four-fold coincidence measurements for the caseof teleportation of the +45◦ polarization state, and graphs c and d show theresults for the +90◦ polarization state. The visibilities, and thus the polariza-tions of the teleported photons, obtained without any background subtractionare 70% ± 3%. These results for teleportation of two non-orthogonal statesprove that we demonstrated teleportation of the quantum state of a singlephoton.


3.4 Discussion

In our experiment, we used pairs of polarization entangled photons as pro-

duced by pulsed down-conversion and two-photon interferometric methods

to transfer the polarization state of one photon onto another one. But, tele-

portation is by no means restricted to this system. In addition to pairs of

entangled photons or entangled atoms [Hagley 1997, Fry 1995], one could

imagine entangling photons with atoms or photons with ions, and so on.

Then teleportation would allow us to transfer the state of, for example, fast-

decohering, short-lived particles onto some more stable systems. This opens

the possibility of quantum memories, where the information of incoming pho-

tons is stored on trapped ions, carefully shielded from the environment.

Furthermore, entanglement purification [Bennett et al., 1996a] is a scheme

of improving the quality of entanglement if it was degraded by decoherence

during storage or transmission of the particles over noisy channels. Then it

becomes possible to send the quantum state of a particle to some place, even if

the available quantum channels are of very poor quality and thus sending the

particle itself would very probably destroy the fragile quantum state. The

feasibility of preserving quantum states in a hostile environment will have

great advantages in the realm of quantum computation. The teleportation

scheme can be used to provide links between quantum computers.

Quantum teleportation is not only an important ingredient in quantum

information tasks; it also allows new types of experiments and investigations

on the foundations of quantum mechanics. As any arbitrary state can be

teleported, so can the fully undetermined state of a particle which is member

of an entangled pair. By doing so, one can transfer the entanglement between

particles. This allows us not only to chain the transmission of quantum

states over distances, where decoherence would have already destroyed the

state completely, but it also enables us to perform a test of Bell’s theorem on

particles which do not share any common past, a new step in the investigation

of the features of quantum mechanics. Last but not least, novel experiments

3.4. DISCUSSION 51

disproving concepts of a the local realistic character of nature become possible

if one uses features of the experiment presented here to generate entanglement

between more than two spatially separated particles [Greenberger et al., 1990;

Zeilinger et al., 1997]. A first such experiment will be reported in Chapter 6.

Since our first experimental realization, quantum teleportation has been

one of the main focus in the field of quantum information physics. In the past

two years, several groups in the world have achieved a number of experimental

advances in quantum teleportation. It is worth comparing these experiments

with our scheme and, briefly discussing the differences among them.

Though Popescu’s suggestion, as realized by De Martini’s group in Rome

[Boschi et al., 1998], provides an elegant way to demonstrate some relevant

Hilbert-space formalism in the absence of a two-photon Bell measurement,

it is not quite proper to claim that the Rome experiment does constitute

teleportation. What really constitutes teleportation? An essential criterion

is to be able to teleport any independent quantum state coming from outside.

This is obviously not possible in the Rome experiment, where the initial

photon has to be entangled from the beginning with the final one. One might

argue that the scheme by Popescu is equivalent to the original teleportation

scheme up to a local operation since, in principle, any unknown state of

a particle from outside could be swapped onto the polarization degree of

freedom of Alice’s EPR particle by a local unitary operation. However, such

a local unitary operation would require a quantum C-NOT gate that does not

exist yet. Further we note, with emphasis, that an experimental realization

of quantum C-NOT gate itself leads a complete Bell measurement and thus

a full realization of the original teleportation scheme, Popescu’s scheme is

therefore not necessary anymore.

Generally speaking, the basic criterions to constitute a bona fide telepor-

tation should be (1) the experimental scheme is capable of teleporting any

state that is designed to teleport, (2) a fidelity better than 2/3 [Massar and

Popescu, 1995], (3) for future applications, at least in principle, the scheme

should be able to be extended to long-distance teleportation, that is, the real


teleportation. Indeed, at the moment only the interference of independently

created photons makes teleportation of any independent and even undefined

(not just simply unknown) photon state possible. Our realization of entan-

glement swapping reported in Chapter 4 will underline the quantum nature

of our teleportation procedures.

The two most recent teleportation experiments were reported by Kim-

ble’s group [Furusawa et al., 1998] and Laflamme’s group [Nielson et al.,

1998], respectively. An obvious advantage of these two schemes is that the

input quantum state can be, in principle, teleported with an efficiency close

to 100%. However, it should be pointed out that while a rather low fidelity

(0.58) was observed in the experiment of Furusawa et al., their scheme is

hardly to be extended to long-distance case because of the unavoidable dis-

persion of squeezed-states during the distribution of entanglement, which

consequently leads to a fast degrading of the quality of squeezed-state entan-

glement. Finally, it is also worthwhile noting that the NMR method used by

Nielson et al. can never teleport a quantum state over macroscopic distance.

Chapter 4

Entanglement Swapping

4.1 Introduction

The phenomenon of entanglement is a remarkable feature of quantum the-

ory. It plays a crucial role in the discussions of the Einstein-Podolsky-Rosen

paradox, of Bell’s inequalities, and of the non-locality of quantum mechanics.

Thus far entanglement has either been realized by having the two entangled

particles emerge from a common source [Freedman 1972, Rarity 1990], or

from having two particles interact either directly or indirectly with each other

[Lamehi-Rachti and Mittig, 1976; Hagley et al., 1997]. Yet, an alternative

possibility to obtain entanglement is to make use of a projection of the state

of two particles onto an entangled state. This projection measurement does

not necessarily require a direct interaction between the two particles: When

each of the particles is entangled with one other partner particle, an appro-

priate measurement, for example, a Bell-state measurement, of the partner

particles will automatically collapse the state of the remaining two particles

into an entangled state. This striking application of the projection postu-

late is referred to as entanglement swapping [Zukowski et al., 1993]. Here,

we report the first experimental realization for entanglement swapping [Pan

1998b]. In our experiment we take two pairs of polarization entangled pho-

tons and subject one photon from each pair to a Bell-state measurement.

53

54 CHAPTER 4. ENTANGLEMENT SWAPPING

This results in projecting the other two outgoing photons into an entangled

state.

4.2 Theoretical scheme

Consider the arrangement of Fig. 4.1. There are two EPR sources, each one

simultaneously emitting a pair of entangled particles. In anticipation to our

experiments we assume that these are polarization entangled photons in the

state

|Ψ〉1234 = 12(|H〉1 |V〉2 − |V〉1 |H〉2)× (|H〉3 |V〉4 − |V〉3 |H〉4)

. (4.1)

Here |H〉 and |V〉 indicate the states of a horizontally and a vertically polar-ized photon, respectively. The total state describes the fact that photons 1

and 2 are entangled in a singlet state in polarization and photons 3 and 4 are

also entangled in the singlet state. Yet, the state of pair 1-2 is factorizable

from the state of pair 3-4, that is, there is no entanglement of any of the

photons 1 or 2 with any of the photons 3 or 4.

We now perform a joint Bell-state measurement on photons 2 and 3, that

is, photons 2 and 3 are projected into one of the four Bell states which form

a complete basis for the combined state of photons 2 and 3

|Ψ±〉23 = 1√2(|H〉2 |V〉3 ± |V〉2 |H〉3)

|Φ±〉23 = 1√2(|H〉2 |H〉3 ± |V〉2 |V〉3) .

(4.2)

This measurement projects photons 1 and 4 also onto a Bell state, a different

one depending on the result of the Bell-state measurement for photons 2 and

3. To consider a specific example let us assume that the result of the Bell-

state measurement of photons 2 and 3 is |Ψ−〉 then it can be seen that the

resulting state for photons 1 and 4 is also |Ψ−〉. In fact, close inspection showsthat for the initial state given in Eq. (4.1) the emerging state of photons 1

4.2. THEORETICAL SCHEME 55

Figure 4.1: Principle of entanglement swapping. Two EPR sources producetwo pairs of entangled photons, pair 1-2 and pair 3-4. One photon fromeach pair (photon 2 and 3) is subjected to a Bell-state measurement (BSM).This results in projecting the other two outgoing photons 1 and 4 into anentangled state. Change of the shading of the lines indicates a change in theset of possible predictions that can be made.

and 4 will be identical to the one photon 2 and 3 collapsed into. This is a

consequence of the fact that the state of Eq. (4.1) can be rewritten as

|Ψ〉1234 = 12(|Ψ+〉14 |Ψ+〉23 − |Ψ−〉14 |Ψ−〉23

− |Φ+〉14 |Φ+〉23 + |Φ−〉14 |Φ−〉23). (4.3)

In all cases photons 1 and 4 emerge entangled despite the fact that they

never interacted in the past. In Fig. 4.1 entangled particles are indicated by

the same line darkness. Note that particles 1 and 4 become entangled after

the Bell-state measurement (BSM) on particles 2 and 3.

The result above can also be interpreted as teleportation of the unknown

state of, say, photon 2 onto photon 4 [Bennett et al., 1993]. In that case

one could consider Alice performing the Bell-state measurement on photons

2 and 3, telling Bob, who is in possession of photon 4, the result of the

Bell-state measurement. Then, by performing one of a fixed set of unitary

operations on photon 4, photons 1 and 4 will be left in an singlet state,


which is exactly the same as the state of photons 1 and 2 before the Bell-

state measurement. It is conceptually most interesting to realize that in this

case the teleported photon state does not have any well-defined polarization,

because it is entangled with photon 1. It is fair to say that here we do not

teleport some unknown state of a photon but rather an in principle undefined

state. The state of photon 2, and therefore also of the teleported photon 4,

is certainly undefined before any measurements are performed on photons 1

or 4.

It is worthwhile noting that the process of entanglement swapping also

gives a means to define that an entangled pair of photons, 1 and 4, is avail-

able. As soon as Alice completes the Bell-state measurement on particles 2

and 3, we know that photons 1 and 4 are on their way ready for detection in

an entangled state. An experimental realization of entanglement swapping

will thus give, for the first time, the possibility to perform a test of Bell’s

inequality using a pair of photons that never interacted. That is a big step

towards the final realization of so called ”event-ready detections” of the en-

tangled particles, a concept suggested by John Bell [Clauser and Shimony,

1978; Bell, 1980].

4.3 Experimental entanglement swapping

As in our teleportation experiment, here we also analyzed only the projection

onto |Ψ−〉23. Figure 4.2 is a schematic drawing of the experimental setup. A

UV laser pulse passing through a BBO crystal creates via type-II parametric

down-conversion the first pair of entangled photons, 1 and 2, in the state

|Ψ−〉12. After reflection the pump pulse passes the crystal again and creates

the second pair of photons, 3 and 4, in the state |Ψ−〉34. Note that the two

pairs are created independently of one another although the same pulse and

the same crystal are used twice. Photon 2 and 3 are subjected to a Bell-

state measurement (BSM). In order to project the state of photons 2 and 3

onto the anti-symmetric Bell-state |Ψ−〉23, the same technique developed for

4.3. EXPERIMENTAL ENTANGLEMENT SWAPPING 57

quantum teleportation has been used in the experiment.

According to the entanglement swapping scheme, upon projection of pho-

ton 2 and 3 into the |Ψ−〉23 state, photon 1 and 4 should be projected into

the |Ψ−〉14 state. To verify that this entangled state is obtained we have to

analyze the polarization correlation between photons 1 and 4 conditioned on

coincidences between the detectors of the Bell-state analyzer. If photon 1

and 4 are in the |Ψ−〉14 state their polarizations should be orthogonal upon

detection in any polarization basis. Using a λ/2 retardation plate at 22.5◦

and two detectors (D1+ and D1−) behind a polarizing beamsplitter we chose

to analyze the polarization of photon 1 both along the +45◦ axis (D1+) and

along the −45◦ axis (D1−). Photon 4 is analyzed by detector D4 at the

variable polarization direction Θ.

If entanglement swapping happens, then both the two-fold coincidences

between D1+ and D4, and between D1− and D4, conditioned on the |Ψ−〉23

detection, should show two sine curves as a function of Θ which are 90◦

out of phase. The D1+D4 curve should, in principle, go to zero for Θ =

45◦ whereas the D1−D4 curve should show a maximum at this position.

Figure 4.3 shows the experimental results for the coincidences between D1+

and D4, and between D1− and D4, given that photons 2 and 3 have been

registered by the two detectors in the Bell-state analyzer. Note that this

method requires four-fold coincidences. The result clearly demonstrates the

expected sine curves, complementary for the two detectors (D1+ and D1−)

of photon 1 registering orthogonal polarizations. We verified by additional

measurements that the sine curves are independent (up to the corresponding

shift in Θ) on the detection basis of photon 1, that is, independent of the

rotation angle of the λ/2 retardation plate. In other words, the observed

sinusoidal behavior of the coincidence rates depends only on the relative

angle between the polarizers in beams 1 and 4.

The experimentally obtained four-fold coincidences have been fitted by a

joint sine function with the same amplitudes for both curves. Note that the

observed visibility of 0.65 clearly surpasses the 0.5 limit of a classical wave


Figure 4.2: Experimental setup. A UV-pulse passing through a non-linearcrystal creates pair 1-2 of entangled photons. Photon 2 is directed to thebeamsplitter (BS). After reflection, during its second passage through thecrystal the UV-pulse creates a second pair 3-4 of entangled photons. Photon3 will also be directed to the beamsplitter to perform a Bell-state measure-ment (BSM) of photons 2 and 3. When photons 2 and 3 yield a coincidenceclick at the two detectors behind the beamsplitter a projecting into the |Ψ−〉23

state takes place. As a consequence of this Bell-state measurement the tworemaining photons 1 and 4 will also be projected onto an entangled state. Toanalyse their entanglement we look at coincidences between detectors D1+

and D4, and between detectors D1− and D4, for different polarization anglesΘ. By rotating the λ/2 plate in front of the polarizing beamsplitter (PBS) wecan also analyze photon 1 in a different orthogonal polarization basis which isnecessary to obtain statements for relative polarization angles between pho-tons 1 and 4. Note that, since the detection of coincidences between detectorsD1+ and D4, and D1− and D4 are conditioned on the detection of the Ψ−

state, we are looking at 4-fold coincidences. Narrow bandwidth filters (F) arepositioned in front of each detector.

4.3. EXPERIMENTAL ENTANGLEMENT SWAPPING 59

theory. A visibility of 0.72±0.04 was observed in a few initial measurements

for analysis along 45o. A future experiment for showing a significant violation

of Bell’s inequalities requires a stable visibility better than 0.71.

In order to find a practical way to improve the visibility of four-photon

interference fringes, let us perform a semi-quantitative estimation of the vis-

ibility via our experimental parameters. If we simply assume that our EPR

pairs are of perfect entanglement quality, i.e. a visibility of 100% for two

entangled photons, then according to Zukowski et al. [Zukowski et al., 1995;

Zeilinger et al., 1997] the visibility of four-photon fringes is given by

Videal =σP√

σ2P + σ2

F

(4.4)

where σP and σF are the spectral width of the pump pulse and the bandwidth

of the interference filters, respectively. Our narrow bandwidth filters σF ≈4.6nm and the measured pulse spectral width σP ≈ 8nm yield Videal ≈ 87%.

However, as observed in the experiment the EPR pairs produced by pulsed

pump have at best a visibility of VEPR ≈ 90% and cannot fully satisfy the re-

quirement of perfect entanglement. Therefore, one must also consider such a

degrading effect on the experimental visibility of four-photon fringes. Because

we observe interference fringes between two independent pairs, one possible

way to estimate the visibility may be the product Videal× V 2EPR. This conse-

quently leads to an estimated visibility of 87%× 90% × 90% ≈ 72%, which

compares favourably with our best experimental results.

Our estimation indicates two possible ways to improve the visibility be-

yond the 0.71 limit. In principle, one could use interference filters with more

narrow bandwidth or try to produce EPR pairs with higher quality of entan-

glement, though both approaches are technically certainly challenging due to

the very low coincidence rate. Therefore, we also expect to improve the very

low four-fold coincidence rate, the main difficulty of the present experiment,

by using a new laser system currently being installed in Vienna, leading to


a better performance of the experiment.

Figure 4.3: Verification of entanglement swapping via verification of entan-glement between the two photons 1 and 4 from separate pairs. Four-fold co-incidences, resulting from two-fold coincidence D1+D4 (D1−D4) conditionedon the Bell-state measurement two-fold coincidences at detectors D2 andD3, as a function of the polarization angle Θ. The two complementary sinecurves with a visibility of 0.65 ± 0.02 demonstrate that photons 1 and 4 arepolarization entangled.

Again, we mention that, obviously, registration of a coincidence in the

two detectors behind the beam splitter could also have been caused by two

pairs created in either source. That possibility could clearly be ruled out by

sophisticated detection procedures. It certainly does not have any implication

on those events in our experiment where we indeed obtain four registration

events.

4.4. GENERALIZATION AND APPLICATIONS 61

4.4 Generalization and applications

While experimental entanglement swapping itself is a further demonstration

of teleportation, i.e. teleportation of quantum mechanically undefined state,

a test of Bell’s theorem using entanglement swapping could test nonlocality

with a pair of particles that never interacted. Such a test would certainly help

us to further understand nonlocality. We could even perform a delayed choice

experiment for entanglement swapping as recently suggested by Peres [Peres,

1999], where one could delay the instant of time of the to perform Bell-state

measurement on photons 2 and 3, and thus entanglement between photons

1 and 4 is produced a posteriori, after they have already been measured and

may no longer exist.

Further, various generalizations of the present scheme are at hand [Bose

1997, Pan 1998a, 1999b]. One could have many different kinds of entan-

glement to begin with, perform various different measurements, and obtain

novel kinds of entanglement for the emerging particles. A first clear possi-

bility [Zukowski et al., 1995; Rarity and Tapster, 1995] is to project three

particles, each from an entangled pair, into a GHZ state [Greenberger et al.,

1990] whenceforth the other three emerging particles are also projected into

a GHZ state.

Secondly, for example, one could use a polarizing beam splitter instead

of the beam splitter in the set-up (Fig. 4.2). Then, using the same reasoning

line as used in section 2.2.2 (or more directly refer to Fig. 2.7 and Eq.(2.33))

we could arrive the following conclusions:

(1) The outgoing four-particle state will be in a conditional GHZ state

immediately after photons 2 and 3 passing through the polarizing beam split-

ter;

(2) as suggested in chapter 2, one could perform ±45◦ polarization anal-ysis at one of the two output ports of the polarizing beam splitter, then

conditioned on the detection of a single photon with 45◦ (or −45◦) polariza-tion, the remaining three photons will be correspondingly projected into a


freely propagating GHZ-state;

(3) one could also perform ±45◦ polarization analysis at the two outputports in the meantime, that is, perform a Bell-state measurement on photons

2 and 3 by using our modified Bell-state analyzer (refer to Fig. 2.5), then by

Eq. (4.3) photons 1 and 4 will be projected into state |Φ+〉14 (or |Φ−〉14) with

respect to a projection of the state of photons 2 and 3 into the Bell-state

|Φ+〉23 (or |Φ−〉23).

The results above exactly imply that one could interpret our experimental

setup in different ways, corresponding to what kind of specific measurement

one intends to perform. It should be noted that such a scheme can be easily

generalized to N particles case by just adding more EPR pairs and polarizing

beam splitters. Again all these schemes above require pulsed pump technol-

ogy, with pulses of even higher power than in the present experiment, yet in

principle achievable with current technology.

We might also remark that the present results, taken together with our

verification of quantum teleportation in chapter 3, are easily understood

in the framework of the Copenhagen interpretation of quantum mechanics

[Nagel, 1989]. They cause no conceptual problems if one accepts that in-

formation about quantum systems is a more basic feature than any possible

”real” properties these systems might have [Zeilinger, 1998].

Finally, it is foreseen that entanglement swapping, besides its interest to

fundamental physics, will have a number of important applications in future

quantum communication schemes. First of all, as mentioned by Bose et al.

[Bose et al., 1998], our entanglement swapping scheme opens up a way to

speed up the distribution of entanglement for any particles possessing mass.

The importance of the distribution of entanglement between distant parties

is obvious as Bell pairs are essential for the implementation of many quantum

communication schemes over large distance, such as secret-key distribution

[Ekert, 1991], teleportation [Bennett et al., 1993] and dense coding [Bennett

and Wiesner, 1992].

4.4. GENERALIZATION AND APPLICATIONS 63

On the other hand, due to the unavoidable decoherence caused by cou-

pling with the environment, the quality of entanglement will be degraded dur-

ing the distribution and storage of entanglement. Therefore, entanglement

purification [Bennett et al., 1996a; 1996b; Deutsch et al., 1996] is of great sig-

nificance to achieve quantum communication with perfect fidelity. Because

all purification schemes involve collective measurements on many photons at

once, or need two-qubit logic gates for polarization-entangled photons, whose

physical implementation is very difficult under the current technology, and

remains to be realized in an experiment. Furthermore, for distances much

larger than the coherence length of a corresponding noisy quantum channel,

the fidelity of transmission is so low that the standard purification methods

above are not applicable. However, while the most recent research [Bose et

al., 1999] shows that a simple variant of our entanglement swapping scheme

can be directly used to purify single pairs of polarization-entangled photons,

it is possible to divide the quantum channel into shorter segments that are

purified separately and then connected by entanglement swapping [Briegel

et al, 1998; Duer et al., 1999]. All this underlines that entanglement swap-

ping is one of the most important key procedures in quantum communication

networks.

Chapter 5

Three-photon GHZentanglement

5.1 Introduction

Ever since the seminal work of Einstein, Podolsky and Rosen [Einstein 1935]

there has been a quest for generating entanglement between quantum par-

ticles. Although two-particle entanglements have long been demonstrated

experimentally [Wu and Shaknov, 1950; Freedman and Clauser, 1972; As-

pect et al., 1982; Kwiat et al., 1995; Hagley et al., 1997], the preparation

of entanglement between three or more particles remains an experimental

challenge. Proposals have been made for experiments with photons [Green-

berger et al., 1990; Zeilinger et al., 1997; Pan and Zeilinger, 1998a] and atoms

[Cirac and Zoller, 1994; Haroche, 1995], and three nuclear spins within a sin-

gle molecule have been prepared such that they locally exhibit three-particle

correlations [Lloyd, 1998; Laflamme et al., 1998]. However, until now there

has been no experiment which demonstrates the existence of entanglement of

more than two spatially separated particles. Here we present the first experi-

mental observation of polarization entanglement of three spatially separated

photons [Bouwmeester, Pan et al., 1999]. Such states, known as Greenberger-

Horne-Zeilinger (GHZ) states, are interesting from both a fundamental and

64

5.2. EXPERIMENTAL SET-UP 65

a technological point of view.

The original motivation to prepare three-particle entanglements stems

from the observation by Greenberger, Horne and Zeilinger that three-particle

entanglement leads to a conflict with local realism for non-statistical predic-

tions of quantum mechanics [Greenberger et al., 1989; 1990; Mermin, 1990a;

1990b]. This is in contrast to the case of Einstein-Podolsky-Rosen experi-

ments with two entangled particles testing Bells inequalities, where the con-

flict only arises for the statistical predictions of quantum theory [Bell, 1964].

The incentive to produce GHZ states has been significantly increased by

the advance of the field of quantum communication and quantum information

processing. Entanglement between several particles is the most important

feature of many such quantum communication and computation protocols

[Bennett, 1995].

5.2 Experimental Set-up

The experiment described here is based on techniques that have been devel-

oped for our previous experiments on quantum teleportation [Bouwmeester,

Pan et al., 1997] and entanglement swapping [Pan et al., 1998]. In fact, one

of the main complications in the those experiments, namely, the creation of

two pairs of photons by a single source, is here turned into a virtue.

The main idea, as was put forward in [Zeilinger et al., 1997; Pan and

Zeilinger, 1998], is to transform two pairs of polarization entangled photons

into a triplet of entangled photons and a fourth independent photon. Our

experimental arrangement is such that we start with two pairs of entangled

photons and register the photons in a way that any information as to which

pair each photon belongs to is erased. Fig. 5.1 is a schematic drawing of our

experimental setup. Pairs of polarization-entangled photons are generated

by a short pulse of ultraviolet (UV) light (≈ 200 fs, λ = 788 nm from a

frequency-doubled, mode-locked Ti-Sapphire laser), which passes through an

66 CHAPTER 5. THREE-PHOTON GHZ ENTANGLEMENT

optically nonlinear crystal (Beta-Barium-Borate, BBO). The probability per

pulse to create a single pair in the desired modes is rather low and of the

order of a few 10−4. The pair creation is such that the following polarization

entangled state is obtained [Kwiat et al., 1995]:

1√2(|H〉a | V 〉b − |V 〉a |H〉b) . (5.1)

This state indicates that there is a superposition of the possibility that the

photon in arm a is horizontally polarized and the one in arm b vertically

polarized (|H〉a |V 〉b), and the opposite possibility, i.e., |V 〉a |H〉b. The mi-nus sign indicates that there is a fixed phase difference of π between the two

possibilities. For our GHZ experiment this phase factor is actually allowed

to have any value, as long as it is the same for both pairs.

The setup is such that arm a continues towards a polarizing beam split-

ter, where V photons are reflected and H photons are transmitted towards

detector T (behind an interference filter δλ = 4.6 nm at 788 nm). Arm b con-

tinues towards a 50/50 polarization-independent beam splitter. From each

beam splitter, one output is directed to a final polarizing beam splitter. In

between the two polarizing beam splitters there is a λ/2 wave plate at an an-

gle of 22.5◦ which rotates the vertical polarization of the photons reflected by

the first polarizing beam splitter into a 45◦ polarization, i.e. a superposition

of |H〉 and |V 〉 with equal amplitudes. The remaining three output arms

continue through interference filters (δλ = 3.6 nm) and single-mode fibers

towards the single-photon detectors D1, D2, and D3. Including filter losses,

coupling into single-mode fibers, and the Si-avalanche detector efficiency, the

total collection and detection probability of a photon is about 10%.

Consider now the case that two pairs are generated by a single UV-pulse,

and that the four photons are all detected, one by each detector T, D1, D2,

and D3. Our claim is that by the coincident detection of four photons and

because of the brief duration of the UV pulse and the narrowness of the

filters, one can conclude that a three-photon GHZ state has been recorded


Figure 5.1: Schematic drawing of the experimental setup for the demonstra-tion of Greenberger-Horne-Zeilinger entanglement for three spatially sepa-rated photons. The UV pulse incident on the BBO crystal produces two pairsof entangled photons. Conditioned on the registration of one photon at thetrigger detector T, the three photons registered at D1, D2, and D3 exhibit thedesired GHZ correlations.


by detectors D1, D2, and D3. The reasoning is as follows. When a four-

fold coincidence recording is obtained, one photon in path a must have been

horizontally polarized and detected by the trigger detector T. Its companion

photon in path b must then be vertically polarized, and it has a 50% chance

to be transmitted by the beam splitter (see Fig. 5.1) towards detector D3

and a 50% chance to be reflected by the beam splitter towards the final

polarizing beam splitter where it will be reflected to D2. Consider the first

possibility, i.e. the companion of the photon detected at T is detected by

D3 and necessarily carried polarization V . Then the counts at detectors D1

and D2 were due to a second pair, one photon traveling via path a and the

other one via path b. The photon traveling via path a must necessarily be V

polarized in order to be reflected by the polarizing beam splitter in path a;

thus its companion, taking path b, must be H polarized and after reflection

at the beam splitter in path b (with a 50% probability) it will be transmitted

by the final polarizing beam splitter and arrive at detector D1. The photon

detected by D2 therefore must be H polarized since it came via path a and

had to transit the last polarizing beam splitter. Note that this latter photon

was V polarized but after passing the λ/2 plate it became polarized at 45◦

which gave it a 50% chance to arrive as an H polarized photon at detector

D2. Thus we conclude that if the photon detected by D3 is the companion of

the T photon, then the coincidence detection by D1, D2, and D3 corresponds

to the detection of the state

|H〉1 |H〉2 |V 〉3 . (5.2)

By a similar argument one can show that if the photon detected by D2 is

the companion of the T photon, the coincidence detection by D1, D2, and D3

corresponds to the detection of the state

|V 〉1 |V 〉2 |H〉3 . (5.3)

In general, the two possible states (5.2) and (5.3) corresponding to a four-

fold coincidence recording will not form a coherent superposition, i.e. a GHZ


state, because they could, in principle, be distinguishable. Besides possible

lack of mode overlap at the detectors, the exact detection time of each photon

can reveal which state is present. For example, state (5.2) is identified by

noting that T and D3, or D1 and D2, fire nearly simultaneously. To erase

this information it is necessary that the coherence time of the photons is

substantially longer than the duration of the UV pulse (approximately 200

fs) [Zukowski et al., 1995]. We achieved this by detecting the photons behind

narrow band-width filters which yield a coherence time of approximately 500

fs. Thus, the possibility to distinguish between states (5.2) and (5.3) is no

longer present, and, by a basic rule of quantum mechanics, the state detected

by a coincidence recording of D1, D2, and D3, conditioned on the trigger T,

is the quantum superposition

1√2(|H〉1 |H〉2 |V 〉3 + | V 〉1 |V 〉2 |H〉3) , (5.4)

which is a GHZ state 1

The plus sign in Eq. (5.4) follows from the following more formal deriva-

tion. Consider two down-conversions producing the product state

1

2(|H〉a |V 〉b − | V 〉a |H〉b)

(|H〉′a |V 〉

′b − |V 〉

′a |H〉

′b

). (5.5)

Initially we assume that the components |H〉a,b and |V 〉a,b created in one

down-conversion might be distinguishable from the components |H〉′a,b and|V 〉′a,b created in the other one. The evolution of the individual componentsof state (5.5) through the apparatus towards the detectors T, D1, D2, and

D3 is given by

|H〉a → |H〉T , (5.6)1Rigorously speaking, this erasure technique is perfect, hence produces a pure GHZ

state, only in the limit of infinitesimal pulse duration and infinitesimal filter bandwidth,but detailed calculations [Zukowski et al., 1995; Horne, 1998] reveal that our pulse andfilter values are sufficient to create a clearly observable entanglement, as confirmed by ourexperimental data.


|V 〉a →1√2(|V 〉1 + |H〉2) , (5.7)

|H〉b →1√2(|H〉1 + |H〉3) , (5.8)

| V 〉b →1√2(|V 〉2 + |V 〉3) . (5.9)

Identical expressions hold for the primed components. Inserting these ex-

pressions into state (5.5) and restricting ourselves to those terms where only

one photon is found in each output we obtain, after normalization

12

{|H〉T

(|V 〉′1 | V 〉2 |H〉

′3 + |H〉

′1 |H〉

′2 | V 〉3

)+ |H〉′T

(|V 〉1 |V 〉

′2 |H〉3 + |H〉1 |H〉2 |V 〉

′3

)} . (5.10)

If now the experiment is performed such that the photon states from the two

down-conversions are indistinguishable, we finally obtain the desired state

(up to an overall minus sign)

1√2|H〉T (|H〉1 |H〉2 |V 〉3 + |V 〉1 |V 〉2 |H〉3) . (5.11)

Note that, even conditioned upon trigger T detecting a single photon,

the total state of the remaining three photons before detection still contains

terms in which, for example, two photons enter the same detector. Thus,

the GHZ entanglement is observed only under the condition that both the

trigger photon and the three entangled photons are actually detected, and in

the following experiments the four-fold coincidence detection actually plays

the double role of both projecting into the desired GHZ state (5.11) and

performing a specific measurement on the state. This, we submit, in practice

will not be a severe limitation because, on the one hand, in any realistic

scheme one always has losses, and information is only obtained if the photons

are actually observed, as, for instance, in third-man quantum cryptography.

On the other hand, many applications explicitly use specific measurement

results. For example, as we will show in chapter 6, the GHZ argument


for testing local realism is based on detection events, and knowledge of the

underlying quantum state is actually not even necessary.

The efficiency for one UV pump pulse to yield such a four-fold coincidence

detection is very low (of the order of 10−10). Fortunately, 7.6×107 UV-pulses

are generated per second, which yields about one double pair creation and

detection per 150 seconds, which is just enough to perform our experiments2. Triple pair creations can be completely neglected since they can give rise

to a four-fold coincidence detection of only very few per day.

Comparing with the scheme suggested in Chapter 2 ( refer to Fig. 2.7 ),

the scheme presented here has a significant advantage for the experimental

alignment. That is that one can easily scan into the region of time overlap of

the photons at the final polarizing beamsplitter simply by observing bunching

effect of two correlated photons generated from a single pump pulse. In

contrast, in the scheme of Fig. 2.7 one has to observe bunching effect of

two photons created by independent sources to find the region of quantum

superposition, this correspondingly decreases the two-fold coincidence rate

by at least three orders of magnitude and thus one has to take much longer

scanning time to see clear two-photon bunching effect.

It is also worth noting that one could simply use a 50:50 beam splitter

instead of the polarizing beam splitter in front of the detector T and take

off the half wave plate λ/2 to observe conditional four-photon GHZ entan-

glement. Following the same reasoning line above, one can easily verify that

the state detected by a four-fold coincidence recording of D1, D2, D3 and T

is in the superposition

1√2(|H〉T |V 〉1 | V 〉2 |H〉3 + |V 〉T |H〉1 |H〉2 | V 〉3) ,

which is a four-photon GHZ state.

2The singles detection rate at detectors D1, D2 and D3, is about 15,000 counts persecond, and at the trigger detector T about 100,000 counts per second, due to the largerfilter bandwith and mode acceptance. Four-fold coincidence is registered with logic ANDcircuitry with a coincidence time of 6 ns.


5.3 Observation of three-photon entanglement

To experimentally demonstrate that a GHZ state has been obtained by the

method described above, we first verified that, conditioned on a photon de-

tection by the trigger T, both the H1H2V3 and the V1V2H3 components

can be observed with the same intensity, but no others. This was done by

comparing the count rates of the eight possible combinations of polarization

measurements, H1H2H3, H1H2V3, ..., V1V2V3. The observed intensity ratio

between the desired and the undesired states was 12:1. Existence of the two

terms as just demonstrated is a necessary but not yet sufficient condition for

demonstrating GHZ entanglement. In fact, there could in principle be just

a statistical mixture of those two states. Therefore, one has to prove that

the two terms coherently superpose. This we did by a measurement of linear

polarization of photon 1 along +45◦, bisecting the H and V direction. Such

a measurement projects photon 1 into the superposition

|+45◦〉1 =1√2(|H〉1 + | V 〉1) , (5.12)

what implies that the state (5.11) is projected into

1√2|H〉T |+45◦〉1 (|H〉2 | V 〉3 + | V 〉2 |H〉3) . (5.13)

Thus photon 2 and 3 end up entangled as predicted under the notion

of ”entangled entanglement” [Krenn and Zeilinger, 1996]. We conclude that

demonstrating the entanglement between photon 2 and 3 confirms the coher-

ent superposition in state (5.11) and thus the existence of the GHZ entangle-

ment. In order to proceed to our experimental demonstration we represent

the entangled state of photons 2 and 3 in a linear basis rotated by 45◦. The

state then becomes

1√2(|+45◦〉2 |+45◦〉3 − |−45◦〉2 | −45◦〉3) , (5.14)

5.3. OBSERVATION OF THREE-PHOTON ENTANGLEMENT 73

which implies that if photon 2 is found to be polarized along -45◦ (or +45◦),

photon 3 is also polarized along the same direction. We test this predic-

tion in our experiment. The absence of the terms |+45◦〉2 | −45◦〉3 and

| −45◦〉2 |+45◦〉3 is due to destructive interference and thus indicates the

desired coherent superposition of the terms in the GHZ state (5.11). The

experiment therefore consisted of measuring four-fold coincidences between

the detector T, detector 1 behind a +45◦ polarizer, detector 2 behind a -45◦

polarizer, and measuring photon 3 behind either a +45◦ polarizer or a -45◦

polarizer. In the experiment, the difference of arrival time of the photons

at the final polarizer, or more specifically, at the detectors D1 and D2, was

varied.

The data points in Fig. 5.2(a) are the experimental results obtained for

the polarization analysis of the photon at D3, conditioned on the trigger and

the detection of two photons polarized at 45◦ and −45◦ by the two detectorsD1 and D2, respectively. The two curves show the four-fold coincidences for a

polarizer oriented at −45◦ (squares) and +45◦ (circles) in front of detector D3

as function of the spatial delay in path a. From the two curves it follows that

for zero delay the polarization of the photon at D3 is oriented along −45◦,in accordance with the quantum-mechanical predictions for the GHZ state.

For non-zero delay, the photons traveling via path a towards the second po-

larizing beam splitter and those traveling via path b become distinguishable.

Therefore increasing the magnitude of delay gradually destroys the quantum

superposition in the three-particle state.

Note that one can equally well conclude from the data that at zero delay,

the photons at D1 and D3 have been projected onto a two-particle entangled

state by the projection of the photon at D2 onto −45◦. The two conclusionsare only compatible for a genuine GHZ state. We note that the observed

visibility was as high as 75%. 3

3The limited visibility is due mainly to the finite width of the interference filters, thefinite pulse duration, and the limited quality of the polarization optics. detector noise oraccidental coincidences do not play any role.


Figure 5.2: Experimental confirmation of GHZ entanglement. Graph (a)shows the results obtained for polarization analysis of the photon at D3, con-ditioned on detection of the trigger photon and detection of one photon atD1 polarized at 45

◦ and one photon at detector D2 polarized −45◦. The twocurves show the four-fold coincidence rates for a polarizer oriented at −45◦and 45◦ respectively in front of detector D3 as a function of the spatial delayin path a. The difference between the two curves at zero delay confirms theGHZ entanglement. By comparison (graph (b)) no such intensity differenceoccurs if the polarizer in front of detector D1 is set at 0

◦. Error bars aregiven by the square root of the coincidence counts.

5.4. DISCUSSION AND CONCLUSION 75

For an additional confirmation of state (5.11) we performed measurements

conditioned on the detection of the photon at D1 under 0◦ polarization (i.e.

V polarization). For the GHZ state (1/√2)(H1H2V3 + V1V2H3) this implies

that the remaining two photons should be in the state V2H3 which cannot give

rise to any correlation between these two photons in the 45◦ detection basis.

The experimental results of these measurement are presented in Fig. 5.2(b).

The data clearly indicate the absence of two-photon correlations and thereby

confirm our claim of the observation of GHZ entanglement between three

spatially separated photons.

5.4 Discussion and conclusion

Although the extension from two to three entangled particles might seem

to be only a modest step forward, the implications are rather profound.

First of all, GHZ entanglements allow for novel tests of quantum mechanics

versus local realistic models [Greenberger et al., 1989; 1990; Mermin, 1990a;

1990b; Zukowski, 1998; Pan et al., 1999a]. Secondly, three-particle GHZ

states might find a direct application, for example, in third-man quantum

cryptography. And thirdly, the method developed to obtain three-particle

entanglement from a source of pairs of entangled particles can be extended to

obtain entanglement between many more particles [Bose et al., 1998], which

is the basis of many quantum communication and computation protocols.

Finally, we note that our experiment, together with our earlier realization

of quantum teleportation [Bouwmeester, Pan et al., 1997] and entanglement

swapping [Pan et al., 1998b] provides the necessary tools to implement a

number of novel entanglement distribution and network ideas as recently

proposed [Grover, 1997; Duer et al., 1999].

Chapter 6

Experimental tests of the GHZtheorem

6.1 Introduction

Ever since its introduction by Schrodinger [Schrodinger, 1935] entanglement

has commanded a central position in the discussions of the interpretation of

quantum mechanics. Originally that discussion has focused on the proposal

by Einstein, Podolsky and Rosen (EPR) of measurements performed on two

spatially separated entangled particles [Einstein et al., 1935]. Most signifi-

cantly, John Bell then showed that there is a conflict between any attempt to

explain the correlations observed in such systems by a local realistic model

and the predictions made by quantum mechanics [Bell, 1964]. In the deriva-

tion of Bell’s inequalities one makes the seemingly innocuous assumption

that perfect correlations can be understood using such a local realistic model

and the conflict then arises for the statistical predictions of quantum theory.

An increasing number of experiments on entangled particle pairs having

confirmed the statistical predictions of quantum mechanics [Freedman et al.,

1972; Aspect et al., 1982; Weihs 1998] have thus provided increasing evidence

against local realistic theories. Yet, one might find some comfort in the fact

76

6.2. THE CONFLICT WITH LOCAL REALISM 77

that such a realistic and thus classical picture can explain perfect correlations

and is only in conflict with statistical predictions of the theory. After all,

quantum mechanics is statistical in its core structure. In other words, for

entangled particle pairs the cases where the result of a measurement on one

particle can definitely be predicted on the basis of a measurement result on

the other particle can be explained by a local realistic model. It is only

that subset of statistical correlations where the measurement results on one

particle can only be predicted with a certain probability which cannot be

explained by such a model.

Yet in 1989 it was shown by Greenberger, Horne and Zeilinger (GHZ) that

for certain three- and four-particle states [Greenberger et al., 1989; 1990] a

conflict with local realism arises even for perfect correlations. That is, even

for those cases where, based on measurement on N − 1 of the particles,

the result of the measurement on particle N can be predicted with certainty.

Local realism and quantum mechanics here both make definite but completely

opposite predictions. A particularly elegant demonstration of that conflict is

due to Mermin [Mermin, 1990a].

Utilizing our recently developed source for three-photon GHZ-entanglements

it is the purpose of this chapter to present a first realization of such a three-

particle test against local realism [Pan et al., 1999a].

6.2 The conflict with local realism

6.2.1 GHZ theorem

How are the quantum predictions of a three-photon GHZ-state in stronger

conflict with local realism than the conflict for two-photon states as implied

78 CHAPTER 6. EXPERIMENTAL TESTS OF THE GHZ THEOREM

by Bell’s inequalities 1? To answer this, consider the state

1√2(|H〉1 |H〉2 |H〉3 + | V 〉1 |V 〉2 |V 〉3) , (6.1)

where H and V denote horizontal and vertical linear polarizations. This

state indicates that the three photons are in a quantum superposition of

the state |H〉1 |H〉2 |H〉3 (the three photons are horizontally polarized) and|V 〉1 | V 〉2 |V 〉3 (the three photons are vertically polarized). We choose this

specific state because it is symmetric with respect to the interchange of all

photons which simplifies the arguments below. The same line of reasoning

holds, however, for any three-particle entangled state.

Consider now some specific predictions following from state (6.1) for mea-

surements of linear polarization along directions rotated by 45◦ with respect

to the original H-V directions, denoted by H ′-V ′, or of circular polarization

denoted by L-R (left-handed, right-handed). These new polarizations can be

expressed in terms of the original ones as

|H ′〉 = 1√2(|H〉+ |V 〉), |V ′〉 = 1√

2(|H〉 − |V 〉) ,(6.2)

|R〉 = 1√2(|H〉+ i |V 〉), |L〉 = 1√

2(|H〉 − i |V 〉) .(6.3)

For convenience we will refer to a measurement of H ′-V ′ linear polarization

as a l measurement and of L-R circular polarization as a c measurement.

Representing state (6.1) in the new states using Eqs. (6.2) and (6.3)

one obtains predictions for measurements of these new polarizations. For

example, for the case of measurement of circular polarization on, say, both

photon 1 and 2, and measurement of linear polarization H ′-V ′ on photon 3,

1For two-photon states Hardy [Hardy, 1993] has found situations where quantum me-chanics predicts a specific result to occur sometimes and local realism predicts the sameresult never to occur [Boschi et al., 1997]


denoted as a ccl experiment, the state may be expressed as

12(|R〉1 |L〉2 |H ′〉3 + |L〉1 |R〉2 |H ′〉3+ |R〉1 |R〉2 |V ′〉3 + |L〉1 |L〉2 | V ′〉3)

(6.4)

This expression has a number of significant implications. Firstly, we note

that any specific result obtained in any individual or in any two-photon

joint measurement is maximally random. For example, photon 1 will exhibit

polarization R or L with the same probability of 50%, or photons 1 and 2

will exhibit polarizations RL, LR, RR, or LL with the same probability of

25%.

Yet secondly we realize that, given any two results of measurements on

any two photons, we can predict with certainty what the result of the corre-

sponding measurement performed on the third photon will be. For example,

suppose photons 1 and 2 are both found to exhibit right-handed (R) circular

polarization. Then by the third term in the expression above, photon 3 will

definitely be V ′ polarized.

By cyclic permutation, we can obtain analogous expressions for any case

of any experiment measuring circular polarization on two photons and H ′-V ′

linear polarization on the remaining one. Thus, in any one of the three ccl,

clc and lcc experiments any individual measurement result both for circular

polarization and for linear H ′-V ′ polarization can be predicted with certainty

for any one of the three photons given the corresponding measurement results

of the other two.

Now we will analyze the implications of these predictions from the point

of view of local realism. First note that the predictions are independent of the

spatial separation of the photons and independent of the relative time order

of the measurements. Let us thus consider the experiment to be performed

such that the three measurements are performed simultaneously in a given

reference frame, say, for conceptual simplicity, in the reference frame of the

source. Thus we can employ the notion of Einstein locality which implies

that no information can travel faster than the speed of light. Hence the


specific measurement result obtained for any photon must not depend on

which specific measurement is performed simultaneously on the other two nor

on the outcome of these measurements. The only way then to explain from a

local realist point of view the perfect correlations discussed above is to assume

that each photon carries elements of reality for both l and c measurements

considered and that these elements of reality determine the specific individual

measurement result [Greenberger et al., 1989; 1990; Mermin, 1990a].

We now consider a fourth experiment measuring linear H ′-V ′ polariza-

tion on all three photons, i.e. a lll experiment. What possible outcomes

will a local realist predict here based on the elements of reality introduced

to explain the earlier ccl, clc and lcc experiments? The state of Eqn. (6.4)

and its permutations imply that whenever we obtain the result H ′ [V ′] for

any one photon, the other two photons must carry elements of reality imply-

ing opposite [identical] circular polarizations. Suppose that for one specific

run of the lll experiment we find, say, the result V ′ both for photon 2 and

for photon 3. Because photon 3 is a V ′, both photon 1 and 2 must carry

identical circular polarization elements of reality; and because photon 2 is a

V ′, both photons 1 and 3 must carry identical circular polarization elements

of reality. This means that all three photons must carry identical circular

polarization elements of reality. Thus, since photons 2 and 3 carry identical

circular polarization elements of reality, photon 1 must necessarily exhibit

linear polarization V ′, it cannot be H ′ polarized. Hence the existence of ele-

ments of reality leads to the conclusion that the result V ′1V′

2V′

3 is one possible

outcome and H ′1V′

2V′

3 is impossible. By parallel constructions, one can verify

that H ′1H′2V′

3 , H ′1V′

2H′3, and V ′1H

′2H′3 are the only other possible outcomes

from a local realistic viewpoint if we elect to measure H ′-V ′ polarizations of

all three particles, i.e. if we perform a lll measurement.

How do these predictions of local realism compare with those of quan-

tum physics? If we express the state given in Eq. (6.1) in terms of H ′-V ′


polarization using Eq. (6.2) we obtain

12(|H ′〉1 |H ′〉2 |H ′〉3 + |H ′〉1 | V ′〉2 | V ′〉3+ |V ′〉1 |H ′〉2 | V ′〉3 + |V ′〉1 |V ′〉2 |H ′〉3) .

(6.5)

Here the local realistic model predicts none of the terms occurring in the

quantum prediction. This implies that whenever local realism predicts a

specific result definitely to occur for a measurement on one of the photons

based on the results for the other two, quantum physics definitely predicts

the opposite result. For example, if two photons are both found to be H ′

polarized, local realism predicts the third photon to carry polarization V ′

while the quantum state predicts H ′.

Thus, while in the case of Bell’s inequalities for two photons the difference

between local realism and quantum physics happens for statistical predictions

of the theory, for three entangled particles the difference occurs already for

the definite predictions, statistics is now only due to inevitable measurement

errors occurring in any and every experiment, even in classical physics.

6.2.2 Generalization to conditional GHZ state

The experiment reported here is based on the observation of three-photon

GHZ entanglement that was achieved in Chapter 5. Conditioned upon that

detector T observes a single photon, the total photon state out of our setup

(Fig. 5.1) actually reads2

|H〉1|V 〉1|V 〉2 + |H〉1|V 〉1|H〉3+|H〉2|V 〉2|H〉1 + |H〉2|V 〉2|V 〉3+|H〉3|V 〉3|V 〉1 + |H〉3|V 〉3|H〉2+|H〉1|H〉2|H〉3 + |V 〉1|V 〉2|V 〉3

(6.6)

2For simplicity of argumentation we have assumed here that for photon 3 H and V aredefined at right angles compared to photons 1 and 2, and also we already exclude the casein which only one EPR pair is produced by a single pulse or two photons enter detectorT. Yet we note that such a simplification does not change the physical conclusion.


which implies that the total photon state produced by our setup, i.e., the state

before detection, also contains terms in which two photons enter the same

detection station. For example, the first two terms of Eq. 6.6 imply there

are two photons in mode 1, and so on. Therefore, the four-fold coincidence

detection here acts as a projection measurement onto the desired GHZ state

(6.1) and filters out those undesirable terms. This might raise doubts about

whether such a source can be used to test local realism.

Actually, the same doubts had been raised earlier for certain Bell-type ex-

periments involving photon pairs [Ou et al., 1988; Shih and Alley, 1988]. Al-

though these experiments have successfully produced the expected quantum-

mechanical correlations, in the past it was often believed [Kwiat et al., 1994,

Caro and Garuccio, 1994] that they could never, not even in their idealized

versions, be consider as genuine tests of local realism. To avoid this problem,

a possible solution is that one could directly prepare unconditional three-

photon GHZ state using the scheme suggested in Chapter 2 (refer to Fig.

2.7).

However, fortunately, Popescu, Hardy and Zukowski [Popescu et al., 1997]

showed that this general belief is wrong and that the above experiments

indeed constitute (modulo the usual detection loopholes) true tests of local

realism. Following the same reasoning line, Zukowski has recently shown

that our GHZ entanglement source enables one to perform a three-particle

test of local realism [Zukowski 1999]. Here we briefly discuss this analysis.

For clarity of the argumentation, let’s first define a local hidden-variable

model for our GHZ entanglement source. In such a model the local events

(i.e., events at one of the observation stations) are determined by the value

of the hidden variables describing the experiment, usually denoted by the

symbol λ, and the local macroscopic controllable parameter set by the local

observer; here the settings of the polarizers in front of detectors D1, D2,

and D3, respectively denoted by x1, x2, and x3. As mentioned already, in

the GHZ argument xi(i = 1, 2, 3) could independently indicate settings for

linear H ′ − V ′ polarization or for circular R − V polarization, i.e. l or c


measurements respectively.

Let Λ be the space of the parameter λ for an ensemble comprised of a

very large number of the states produced by our GHZ entanglement source.

From Eq. 6.6, it is easy to see that such an ensemble can be partitioned into

two disjoint sub-ensembles Λa and Λb, i.e. corresponding to those for which

(a) one photon each is detected in each of the outgoing beams 1, 2 and 3,

(b) two photons are observed in one of the three outgoing beams, and one

photon is observed in one of the remaining two beams. Denoting by ρ(λ)

the distribution of the hidden variable λ, then according to local realism, the

distribution ρ(λ) of the union of the two subsets is clearly independent of

the settings of the polarizers in front of detectors D1, D2, and D3. However,

it is not evident whether the mode of partitioning is also independent of the

settings of the polarizers. If one wants to construct the GHZ argument within

the sub-ensemble Λa, one must thus demonstrate that the distribution ρa(λ)

of the sub-ensemble Λa is also independent of the polarizer settings. For a

thorough discussion of the essence of that argument, refer to [Clauser and

Shimony, 1978] and references therein.

To do so, imagine that detectors D1 and D2 each detects a single photon

(under two specified polarizer settings, x1, x2 respectively), then there is

definitely a single photon to be detected by D3, no matter what the polarizer

setting is there3. This implies that, (1) the occurrence of such a joint event,

i.e. whether a joint event belongs to the subset Λa or not, is irrespective of

the local polarizer setting of D3; (2) the local occurrence of a single photon

detection by D3, which belongs to the subset Λa, is also independent of the

polarizer setting of D3.

By parallel reasoning, one can finally conclude that the occurrence of a

joint event belonging to the sub-ensemble Λa depends only on the hidden

variable λ and not upon the local settings of the polarizers. Furthermore, in

3 Because the four photons were produced by a single pulse, then conditioned upon thatdetectors D1, D2 and T each detects a single photon, detector D3 must also correspondinglydetect a single photon. Here, we always suppose the detectors have perfect 100% detectionefficiency.


the subset the occurrence of a single photon detection at any local observation

station also does not depend on the polarizer settings at D1, D2, and D3.

These exactly confirm that the distribution ρa(λ) of the sub-ensemble Λa

is independent of the polarizer orientations, by which we could define the

probability measures only on the subset Λa and take ρa(λ) to have norm one:

∫Λa

dρa = 1 (6.7)

Thus, by limiting ourselves only to the sub-ensemble Λa we can predict the

possible outcomes in a lll experiment via local realism, then compare with the

quantum mechanical prediction. Therefore, we finally arrive at the following

conclusion:

In essence, the GHZ argument for testing local realism is based on de-

tection events, and knowledge of the underlying quantum state is not even

necessary. It is indeed enough to consider only those four-fold coincidences

discussed above and ignore totally the contributions by the other terms.

6.3 Experimental results

As explained in section 6.2.1 demonstration of the conflict between local

realism and quantum mechanics for GHZ entanglement consists of four ex-

periments each with three spatially separated polarization measurements.

First, one performs ccl, clc, and lcc experiments. If the results obtained are

in agreement with the predictions for a GHZ state then the predictions for

an lll experiment are exactly opposite for a local realist theory as to that of

quantum mechanics.

For each experiment we have 8 possible outcomes of which ideally 4 should

never occur. Obviously, no experiment neither in classical physics nor in

quantum mechanics can ever be perfect and therefore, due to principally un-

avoidable experimental errors, even the outcomes which should not occur will

6.3. EXPERIMENTAL RESULTS 85

occur with some small probability in any realistic experiment. The question

is how to deal with this problem in view of the fact that the GHZ argument

is based on perfect correlations.

In the present chapter we follow two independent possible strategies. In

the first strategy we simply compare our experimental results with the pre-

dictions both of quantum mechanics and of a local realist theory for GHZ

correlations assuming that the particles carry the hidden variables necessary

to explain the perfect quantum ccl, clc and lcc correlations. The spurious

events are then just due to experimental imperfection not correlated to the

hidden variables a photon carries. A local realist might argue against that

approach and suggest that the non-perfect detection events indicate that the

GHZ argumentation cannot succeed. In our second strategy we therefore

give maximum leeway to local realist theories assuming that the non-perfect

events in the first three experiments indicate a set of hidden variables (el-

ements of reality) which are in extreme conflict with quantum mechanics.

We then compare the local realist prediction for the lll experiment obtained

under that assumption with the experimental results.

In the experiment, all measurements are conditioned upon the detection

of a photon by the trigger detector T and we will only refer to the remaining

three photons which are detected by detectors D1, D2, and D3. To meet the

condition of time overlap of the photons at the final polarizing beamsplitter

PBS (Fig. 5.1) we change in small steps the time difference between the pho-

tons from arm a and b by translating the position of the beamsplitter BS.

In this way, we can scan into the region of quantum superposition. Polar-

izers and λ/4 plates have been used to perform polarization analysis. More

specifically, we insert a polarizer oriented at 45◦ or -45◦ in front of a certain

detector to perform a H ′ or V ′ polarization measurement respectively, and

further insert a λ/4 plate in front of the polarizer to perform a R or L circular

polarization measurement.

The observed results for two possible outcomes in a ccl experiment are

shown in Fig. 6.1(a). The remaining possible outcomes of a ccl experiment


have also been measured. At large delay, i.e. outside the region of coherent

superposition, it was observed that within the experimental accuracy the

eight possible outcomes have the same coincidence rate, whose mean value

was chosen as a normalization standard. After normalizing we determined

the fractions for all eight possible outcomes simply by dividing the normal-

ized four-fold coincidences of a specific outcome by the sum of all possible

outcomes in a ccl experiment. For instance the two black bars in Fig. 6.1(b)

indicate the fractions of the RRV ′ and RRH ′ contributions at zero delay

respectively.

All individual fractions which were obtained in our ccl, clc and lcc exper-

iments are shown in Figs. 6.2(a), (b) and (c), respectively. From the data we

conclude that we observe the GHZ terms of Eq. 6.4 predicted by quantum

mechanics in 85% of all cases and in 15% we observe spurious events.

Adopting our first strategy we assume the spurious events are just due to

experimental errors and thus conclude within the experimental accuracy that

for each photon 1, 2 and 3, quantities corresponding to both c and l mea-

surements are elements of reality. Consequently a local realist if he accepts

that reasoning would thus predict that for a lll experiment, the combinations

V ′V ′V ′, H ′H ′V ′, H ′V ′H ′, and V ′H ′H ′ will only be observable (Fig. 6.3(b)).

However referring back to our original discussion we see that quantum me-

chanics predicts the exact opposite terms should be observed (Fig. 6.3(a)).

To settle this conflict we then perform the actual lll experiment. Our results,

shown in Fig. 6.3(c), disagree with the local realism predictions and are con-

sistent with the quantum mechanical predictions. The individual fractions

in Fig. 6.3(c) clearly show within our experimental uncertainty that only

those triple coincidences predicted by quantum mechanics occur and not

those predicted by local realism. In this sense, we claim that we experimen-

tally realized the first three-particle test of local realism following the GHZ

argument.

We have already seen that the observed results for a lll experiment con-

firm the quantum mechanical predictions when we assume that deviations


Figure 6.1: A typical experimental result used in the GHZ argument, in thiscase four-fold coincidences (top) between the trigger detector T, detectors D1

and D2 both set to measure a right-handed polarized photon, and detector D3

set to measure a linearly polarized H ′ (lower) and V ′ (upper curve) photon asa function of the delay between photon 1 and 2 at the final polarizing beam-splitter. At zero delay maximal GHZ entanglement results and the bottomgraph shows the experimentally determined fractions of RRV ′ and RRH ′

triples (out of the eight possible outcomes in the ccl experiment) as deducedfor the zero delay measurements


Figure 6.2: Fractions of the various outcomes observed in the ccl, clc, andlcc experiments. The experimental data show that we observe the GHZ termspredicted by quantum physics in (85± 4)% of all cases and in (15± 2)% the


Figure 6.3: The conflicting predictions of quantum physics (a) and local real-ism (b) of the fractions of the various outcomes in a lll experiment for perfectcorrelations. The experimental results (c) are in agreement with quantumphysics within experimental errors and in disagreement with local realism.


from perfect correlations in our experiment, and in any experiment for that

matter, are just due to unavoidable experimental errors. However what are

the predictions of local realism for the lll experiment when the correlations

are not perfect, as is the case for our experiment where not all events observed

in the ccl, clc and lcc experiments agree with the quantum GHZ predictions?

Is it possible that by using a local realistic theory, these non-GHZ terms can

explain all our experimental results?

To answer this we adopt our second strategy and consider the best predic-

tion that a local realistic theory could obtain using these spurious terms. How

could, for example, a local realist obtain the quantum prediction H ′H ′H ′?

One possibility is to assume that triple events producing H ′H ′H ′ would be

described by a specific set of local hidden variables such that they would give

events in agreement with quantum theory both in a lcc and clc experiment,

for example the results H ′LR and LH ′R, but a spurious event for a ccl ex-

periment, namely LLH ′. In this way any local realistic prediction for an

event predicted by quantum theory in our lll experiment will use at least one

spurious event in the earlier measurements together with two correct ones.

Therefore the fraction of quantum predictions in the lll experiment can at

most be equal to the sum of the fractions of all spurious events in the ccl,

clc, and lcc experiments, that is 0.45. However, we experimentally observed

such terms with a fraction of 0.87± 0.04 (Fig. 6.3(c)), in clear contradictionto the hidden variable prediction.

6.4 Discussion and Prospects

Since the first tests of quantum mechanics versus local realism there have

been strong debates as to what extent these experiments fully refute the

notion of local realism. In this chapter we presented the first experimental

test of quantum nonlocality in three-particle entanglement where the theories

make definite but opposite predictions. Our experiment fully confirms the

predictions of quantum mechanics and is in conflict with local hidden variable

6.4. DISCUSSION AND PROSPECTS 91

theories. We would like to remark that our second analysis presented above,

succeeds because our average visibility of (71 ± 4)% clearly surpasses the

minimum of 50% necessary for a violation of local realism [Mermin, 1990b;

Roy and Singh, 1991; Zukowski and Kaszlikowski, 1997; Ryff, 1997].

However, we have by no means the illusion that our new test will once

and for all convince the disbelievers of quantum mechanics. Our experiment

shares with all existing two-particle tests of local realism the property that

the detection efficiencies are rather low. Therefore we had to invoke the fair

sampling hypothesis [Pearle, 1970; Clauser and Shimony, 1978] where it is

assumed that the registered events are a faithful representative of the whole

ensemble.

It will be interesting to further study GHZ correlations over large dis-

tances with space-like separated randomly switched measurements [Weihs

et al., 1998], to extend the techniques used here to the observation of multi-

photon entanglement [Bose et al., 1998], to observe GHZ-correlations in mas-

sive objects like atoms [Hagley et al., 1997], and to investigate possible ap-

plications in quantum computation and quantum communication protocols

[Briegel et al., 1998; Cleve and Buhrman, 1997].

Chapter 7

Conclusions and outlook

In this work, we have used pairs of polarization-entangled photons as pro-

duced by pulsed parametric down-conversion to experimentally explore in-

terference phenomena of multiparticle quantum systems. Our research has

been mainly concentrated on the experimental demonstration of quantum

teleportation, on the experimental realization of entanglement swapping, on

the production of three-particle GHZ entanglement, and on the experimental

realization of a three-particle test of local reality versus quantum mechan-

ics. For the first time, these experiments open the door to study various

novel phenomena for quantum systems of three or more particles. It is fore-

seen that the techniques developed in our experiments, besides their interest

to the foundations of quantum physics, will have many important applica-

tions in future quantum communication schemes, such as third-man quantum

cryptography, entanglement purification, and entanglement distribution.

Although teleportation has been realized using polarization-entangled

photons, we are still on the way to long-distance quantum teleportation,

i.e. transmission of quantum states over large distance. Utilizing the tools

that were recently developed for those long-distance Bell-type experiments

[Weihs et al., 1998; Tittel et al., 1998a; 1998b], a slight modification of our

teleportation scheme will allow us to realize long-distance teleportation with

an efficiency of 50%. As discussed in Chapter 3, a stable visibility better than

92

93

71% is necessary to violate a Bell inequality. It is of great interest to inves-

tigate the possibility to improve the visibility of our entanglement swapping

experiment, this will ultimately leads to an experimental test of nonlocality

both with a pair of photons that never interacted and truly independent ob-

servers. It is proposed that one can produce four-photon GHZ state [Pan

and Zeilinger, 1999b]. The long-distance GHZ experiment will constitute

a test for local realism under strict Einstein locality condition. It is also

suggested to perform long-distance quantum cryptography based on multi-

particle (GHZ) entanglement, which enables a more advanced cryptography

system.

We have seen in the thesis that our interferometric Bell-state analyzer

does not give the full capacity of the new quantum communication schemes.

Three instead of four messages can be encoded in one photon and the tele-

portation of polarization states of photon can be performed only with a

maximum of 50% efficiency. To perform all possible unitary transforma-

tions strong coupling between quantum systems is necessary. We propose

to continue the development of photon-photon coupling in optical systems.

As suggested by Roch et al. [Roch et al., 1992], the experiments recently

performed by Kimble’s group at Caltech [Turchette et al., 1995, Hood et

al., 1998] show a way to increase the coupling between weak light beams

by atoms in high finesse cavities. However, the bandwidth requirements of

these devices are too high to combine this approach with down-conversion

experiments.

It will be a real future challenge to study similar systems with slightly

relaxed bandwidth requirements, in order to adapt the technique for our

experiments. This would ultimately lead to the realization of complete long-

distance teleportation of quantum states of photon and atom. Moreover,

this would also open a wholly new field of experimental investigations, since

such photon-photon-coupling devices are needed for quantum nondemolition

measurements, quantum logic gates, and in various experiments on the foun-

dations of quantum mechanics.

Bibliography

[Aspect et al., 1982] A. Aspect, P. Grangier, and G. Roger. Experimental

test of Bell’s inequalities using time-varying analyzers. Phys. Rev. Lett.,

49, 1804-1807, 1982.

[Barenco et al., 1995a] A. Barenco, D. Deutsch, A. Ekert, and R. Jozsa. Con-

ditional quantum dynamics and logic getes. Phys. Rev. Lett., 74, 4083-

4086, 1995.

[Barenco et al., 1995b] A. Barenco, C.H. Bennett, D.P. DiVincenzo, N. Mar-

golus, P. Shor, T. Sleator, J. Smolin, and H. Weinfurter. Phys. Rev.

A52, 3457-3467.

[Bell, 1964] J.S. Bell. On the Einstein Poldolsky Rosen paradox. Physics, 1,

195-200, 1964.

[Bennett et al., 1992] C.H. Bennett, G. Brassard, and A. Ekert. Quantum

Cryptography. Sci. Am., 257, 50-57, 1992.

[Bennett and Wiesner, 1992] C.H. Bennett, and S. Wiesner. Communication

via one- and two-particle operators on Einstein-Podolsky-Rosen states.

Phys. Rev. Lett., 69, 2881-2884, 1992.

[Bennett et al., 1993] C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A.

Peres, and W.K. Wootters. Teleporting an unknown quantum state via

dual classic and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 70,

1895-1899, 1993.

94

BIBLIOGRAPHY 95

[Bennett, 1995] C.H. Bennett. Quantum information and computation.

Phys. Today 48, 24-30, October, 1995.

[Bennett et al., 1996a] C.H. Bennett, G. Brassard, S. Popescu, B. Schu-

macher, J.A. Smolin, W.K. Wootters. Purification of noisy entangle-

ment and faithful teleportation via noisy channels. Phys. Rev. Lett. 76,

722-725, 1996.

[Bennett et al., 1996b] C.H. Bennett, H.J. Bernstein, S. Popescu, and B.

Schumacher. Concentrating partial entanglment by local operations.

Phys. Rev. A 53, 2046-2052, 1996.

[Boschi et al., 1997] D. Boschi, S. Branca, F. De Martini, and L. Hardy.

Ladder Proof of Nonlocality without Inequalities: Theoretical and Ex-

perimental Results. Phys. Rev. Lett. 79, 2755-2758, 1997.

[Boschi et al., 1998] D. Boschi, S. Branca, F. De Martini, L. Hardy, and S.

Popescu. Experimental Realization of Teleporting an Unknown Pure

Quantum State via Dual Classical and Einstein-Podolsky-Rosen Chan-

nels. Phys. Rev. Lett. 80, 1121-1125, 1998.

[Bose et al., 1998] S. Bose, V. Vedral, and P.L. Knight. Multiparticle gener-

alization of entanglement swapping. Phys. Rev. A 57, 822-829, 1998.

[Bose et al., 1999] S. Bose, V. Vedral, and P.L. Knight. Purification via en-

tanglement swapping and conserved entanglement. Phys. Rev. A 60,

194-197, 1999.

[Bouwmeester, Pan et al., 1997] D. Bouwmeester, J.W. Pan, K. Mattle, M.

Eibl, H. Weinfurter, and A. Zeilinger. Experimental quantum telepor-

tation. Nature(London) 390, 575-579, 1997.

[Bouwmeester, Pan et al., 1999] D. Bouwmeester, J. W. Pan, M. Daniell, H.

Weinfurter, and A. Zeilinger. Observation of three-photon Greenberger-

Horne-Zeilinger Entanglement. Phys. Rev. Lett. 82, 1345-1349, 1999.

96 BIBLIOGRAPHY

[Braustein and Mann, 1995] S.L. Braunstein, and A. Mann. Measurement of

the Bell operator and quantum teleportation. Phys. Rev. A 51, R1727-

R1730, 1995.

[Briegel et al., 1998] H.-J. Briegel, W. Duer, J.I. cirac, and P. Zoller. Quan-

tum repeaters: The role of imperfect local operations in quantum com-

munication. Phys. Rev. Lett. 81, 5932-5935, 1998.

[Bruss et al., 1997] D. Bruss, A. Ekert, S.F. Huelga, J.W. Pan, and A.

Zeilinger. Quantum computing with controlled-NOT and few qubits.

Phil. Trans. R. Soc. Lond. A 355, 2259-2266, 1997.

[Carnal and Mlynek, 1991] O. Carnal and J. Mlynek. Young’s double-slit ex-

periment with atoms: A simple atom interferometer. Phys. Rev. Lett.

66, 2689-2692, 1991.

[Caro and Garuccio, 1994] L. De Caro, and A. Garuccio. Reliability of Bell-

inequality measurements using polarization correlations in parametric-

down-conversion photons sources. Phys. Rev. A 50, R2803-R2805,

1994.

[Chuang et al., 1998] I.L. Chuang, L.M.K. Vandersypen, X.L. Zhou, D.W.

Leung, and S. Lloyd. Experimental realization of a quantum algorithm.

Nature(London) 393, 143-146, 1998.

[Clauser and Shimony, 1978] J. Clauser, and A. Shimony. Bell’s theorem:

experimental tests and implications. Rep. Prog. Phys. 41, 1881-1927,

1978.

[Cleve and Buhrman, 1997] R. Cleve and H. Buhrman. Phys. Rev. A 56,

1201, 1997.

[Cirac and Zoller, 1994] J.I. Cirac, and P. Zoller. Preparation of macroscopic

superposition in many-atom systems. Phys. Rev. A 50, R2799-R2802,

1994.

BIBLIOGRAPHY 97

[Deutsch, 1985] D. Deutsch. Quantum theory, the church-Turing principle

and the universal quantum computer. Proc. R. Soc. Lond. A 400, 97-

115, 1985.

[Deutsch et al., 1996] D. Deutsch, A. Ekert, R. Jozsa, C. macchiavello, S.

Popescu, and A. Sanpera. Quantum privacy amplification and the se-

curity of quantum cryptography over noisy channels. Phys. Rev. Lett.

77, 2818-2821, 1996.

[Duer et al., 1999] W. Duer, H.-J. Briegel, J.I. Cirac, and P. Zoller. Quantum

repeaters based on entanglement purification. Phys. Rev. A 59, 169-

181, 1999.

[Einstein et al., 1935] A. Einstein, B. Podolsky, N. Rosen. Can quantum-

mechanical description of physical reality be considered complete?

Phys. Rev. 47, 777-779, 1935.

[Ekert, 1991] A.K. Ekert. Quantum cryptography based on Bell’s theorem.

Phys. Rev. Lett. 67, 661-663, 1991.

[Freedman and Clauser 1972] S.J. Freedman, and J.F. Clauser. Experimen-

tal test of local hidden-variable theories. Phys. Rev. Lett. 28, 938-941,

1972.

[Fry et al., 1995] E.S. Fry, T. Walther, and S. Li. Proposal for a loophole-free

test of the Bell inequalities. Phys. Rev. A 52, 4381-4395, 1995.

[Furusawa et al., 1998] A. Furusawa, J.L. Sorensen, S.L. Braunstein, C.A.

Fuchs, H.J. Kimble, E.S. Polzik. Unconditional quantum teleportation.

Science 282, 706-709, 1998.

[Greenberger et al., 1989] D.M. Greenberger, M.A. Horne, A. Zeilinger. Go-

ing beyond Bell’s theorem. in Bell’s Theorem, Quantum Theory, and

Conceptions of the Universe, edited by M. Kafatos, (Kluwer Aca-

demics, Dordrecht, The Netherlands, 1989), pp. 73-76.

98 BIBLIOGRAPHY

[Greenberger et al., 1990] D.M. Greenberger, M.A. Horne, A. Shimony, A.

Zeilinger. Bell’s theorem without inequalities. Am. J. Phys. 58, 1131-

1143, 1990.

[Greenberger et al., 1993] D.M. Greenberger, M.A. Horne, A. Zeilinger. Mul-

tiparticle interferometry and the superposition principle. Phys. Today

46, 22-29, August 1993.

[Grover, 1997] L.K. Grover. Quantum telecomputation. quant-ph/9704012.

[Hagley et al., 1997] E. Hagley, X. Maitre, G. Nogues, C. Wunderlich, M.

Brune, J.M. Raimond, S. Haroche. Generation of Einstein-Podolsky-

Rosen pairs of atoms. Phys. Rev. Lett. 79, 1-5, 1997.

[Hardy, 1993] L. Hardy. Nonlocality for two particles withouts inequalities

for almost all entangled states. Phys. Rev. Lett. 71, 1665-1668, 1993.

[Haroche, 1995] S. Haroche. Atoms and photons in high-Q cavities: next

tests of quantum theory. Ann. N. Y. Acad. Sci. 755, 73-86, 1995.

[Horne, 1998] M. A. Horne. Fortschr. Phys. 46, 6, 1998.

[Hood et al., 1998] C.J. Hood, M.S. Chapman, T.W. Lynn, and H.J. Kimble.

Real-time cavity-QED with single atoms. Phys. Rev. Lett. 80, 4157-

4160, 1998.

[Keith et al., 1991] D.W. Keith, C.R. Ekstrom, Q.A. Turchette, and D.E.

Pritchard. An interferometer for atoms. Phys. Rev. Lett. 66, 2693-2696,

1991.

[Krenn and Zeilinger, 1996] G. Krenn, and A. Zeilinger. Entangled entangle-

ment. Phys. Rev. A 54, 1793-1796, 1996.

[Kwiat et al., 1994] P.G. Kwiat, P.E. Eberhard, A.M. Steinberger, and R.Y.

Chiao. Proposal for a loophole-free Bell inequality experiment. Phys.

Rev. A 49, 3209-3220, 1994.

BIBLIOGRAPHY 99

[Kwiat et al., 1995] P.G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A.V.

Sergienko, and Y.H. Shih. New high intensity source of polarization-

entangled photon pairs. Phys. Rev. Lett. 75, 4337-4341, 1995.

[Laflamme et al., 1998] R. Laflamme, E. Knill, W.H. Zurek, P. Catasti, and

S.V.S. Mariappen. NMR Greenberger-Horne -Zeilinger states. Phil.

Trans. R. Soc. Lond. A 356, 1941-1947, 1998.

[Lamehi-Rachti and Mittig, 1976] M. Lamehi-Rachti, and W. Mittig. Quan-

tum mechanics and hidden variables: A test of Bell’s inequality by

the measurement of the spin correlation in low-energy proton-proton

scattering. Phys. Rev. D 14, 2543-2555, 1998.

[Lloyd, 1998] S. Lloyd. Microscopic analogs of the Greenberger-Horne-

Zeilinger experiment. Phys. Rev. A 57, R1473-1476, 1998.

[Marton et al., 1954] L. Marton, J.A. Simpson, and J.A. Suddeth. Phys. Rev.

90, 490, 1954.

[Massar and Popescu, 1995] S. Massar, and S. Popescu. Optimal extraction

of information from infinite quantum ensembles. Phys. Rev. Lett. 74,

1259-1262, 1995.

[Mattle et al., 1996] K. Mattle, H. Weinfurter, P.G. Kwiat, and A. Zeilinger.

Dense coding in experimental quantum communication. Phys. Rev.

Lett. 76, 4656-4659, 1996.

[Mermin, 1990a] N.D. Mermin. What’s wrong with these elements of reality?

Phys. Today 43, 9-11, June 1990.

[Mermin, 1990b] N.D. Mermin. Extreme quantum entanglement in a super-

position of macroscopically distinct states. Phys. Rev. Lett. 65, 1838-

1841, 1990.

[Nagel, 1989] B. Nagel. in Possible Worlds in Humanities, Arts and Sciences,

Ed. by Allen, Proceedings of Nobel Symposium 65 (Gruyter, Berlin,

1989), p. 388.

100 BIBLIOGRAPHY

[Nielsen et al., 1998] M.A. Nielsen, E. Knill, and R. Laflamme. Com-

plete quantum teleportation using nuclear magnetic resonance. Na-

ture(London) 396, 52-55, 1998.

[Ou et al., 1988] Z.Y. Ou, and L. Mandel. Violation of Bell’s inequality and

classical probability in a two-photon correlation experiment. Phys. Rev.

Lett. 61, 50-53, 1988.

[Pan and Zeilinger, 1998] J.W. Pan, and A. Zeilinger. Greenberger-Horne-

Zeilinger-state analyzer. Phys. Rev. A 57, 2208-2211, 1998.

[Pan et al., 1998] J.W. Pan, D. Bouwmeester, H. Weinfurter, and A.

Zeilinger. Experimental entanglement swapping: entangling photons

that never interacted. Phys. Rev. Lett. 80, 3891-3894, 1998.

[Pan et al., 1999] J.W. Pan, D. Bouwmeester, M. Daniell, H. Weinfurter, and

A. Zeilinger. Experimental test of quantum nonlocality in three-photon

GHZ entanglement. submitted to Nature, 1999.

[Pan and Zeilinger, 1999] J.W. Pan, and A. Zeilinger. Practical Scheme for

the Production of Multi-photon Entanglement. in preparation, 1999.

[Pearle, 1970] P. Pearle. Phys. Rev. D 2, 1418, 1970

[Peres, 1999] A. Peres. Delayed choice for entanglement swapping. to appear

in a special issue of Journal of Modern Optics, 1999.

[Popescu et al., 1997] S. Popescu, L. Hardy, and M. Zukowski. Revisiting

Bell’s theorem for a class of down-conversion experiments. Phys. Rev.

A 56, R4353-4357, 1997.

[Rarity and Tapster, 1990] J.G. Rarity, and P.R. Tapster. Experimental vi-

olation of Bell’s inequality based on phase and momentum. Phys. Rev.

Lett. 64, 2495-2498, 1990.

[Rarity, 1995] J.G. Rarity. Interference of single photons from separate

sources. Ann. N. Y. Acad. Sci. 755, 624-631, 1995.

BIBLIOGRAPHY 101

[Rauch et al., 1974] H. Rauch, W. Treimar, and U. Bonse. Test of a Single

Crystal Neutron Interferometer. Phys. Lett. A 57, 369, 1974.

[Ryff, 1997] L.C. Ryff. Am. J. Phys. 65, 1197, 1997.

[Roch et al., 1992] J.F. Roch, G. Roger, P. Grangier, J. Courty, and S. Rey-

naud. Appl. Phys. B 55, 291, 1992, and references therein.

[Roy and Singh, 1991] S.M. Roy, and V. Singh. Tests of signal locality and

Einstein-Bell locality for Multiparticle systems. Phys. Rev. Lett. 67,

2761-2764, (1995).

[Rubin et al., 1994] M.H. Rubin, D.N. Klyshko, Y.H. Shih, and A.V.

Sergienko. Theory of two-photon entanglement in type-II optical para-

metric down-conversion. Phys. Rev. A50, 5122-5133, 1995.

[Schrodinger, 1935] E. Schrodinger. Die gegenwartige Situation in der Quan-

tenmechanik. Naturwissenschaften 23, 807-812; 823-828; 844-849,

1935.

[Schumacher, 1995] B. Schumacher. Quantum coding. Phys. Rev. A 51,

2738-2747 (1995).

[Shih and Alley, 1988] Y.H. Shih, and C.O. Alley. New type of Einstein-

Podolsky-Rosen-Bohm experiment using pairs of light quanta produced

by optical parametric down conversion. Phys. Rev. Lett. 61, 2921-2924,

1988.

[Shor, 1994] P.W. Shor. Algorithms for quantum computation: Discrete log-

arithms and factoring. in Proceedings of the 35th Annual Symposium on

Foundations of Computer Science. Ed. by Goldwasser, IEEE Computer

Society Press, Los Alamitos, 1994, pp. 124-134.

[Tittel et al., 1998a] W. Tittel, J. Brendel, B. Gisin, T. Herzog, H. Zbinden,

and N. Gisin. Experimental demonstration of quantum correlations

over more than 10 km. Phys. Rev. A 57, 3229-3232, 1998.

102 BIBLIOGRAPHY

[Tittel et al., 1998b] W. Tittel, J. Brendel, H. Zbinden, and N. Gisin. Viola-

tion of Bell inequalities by photons more than 10km apart. Phys. Rev.

Lett. 81, 3563-3566, 1998.

[Turchette 1995] Q.A. Turchette, C.J. Hood, W. Lange, H. Mabuchi, and

H.J. Kimble. Measurement of conditional phase shifts for quantum

logic. Phys. Rev. Lett. 75, 4710-4713, 1995.

[Weihs et al., 1998] G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and

A. Zeilinger. Violation of Bell’s inequality under strict Einstein locality

conditions. Phys. Rev. Lett. 81, 5039-5043, 1998.

[Weinfurter, 1994] H. Weinfurter. Experimental Bell-state analysis. Euro-

phys. Lett. 25, 559-564, 1994.

[Wootters and Zurek, 1982] W.K. Wootters, and W.H. Zurek. A single quan-

tum cannot be cloned. Nature(London) 299, 802-803, 1982.

[Wu and Shaknov, 1950] C.S. Wu, and I. Shaknov. The angular correlation

of scattered annihilation radiation. Phys. Rev. 77, 136, 1950.

[Zeilinger, 1981] A. Zeilinger. General properties of lossless beam splitters in

interferometry. Am. J. Phys. 49, 882, 1981.

[Zeilinger et al., 1997] A. Zeilinger, M.A. Horne, H. Weinfurter, and M.

Zukowski. Three particle entanglements from two entangled pairs.

Phys. Rev. Lett. 78, 3031-3034, 1997.

[Zeilinger, 1998] A. Zeilinger. Quantum entanglement: A fundamental con-

cept finding its applications. Phys. Scr. T76, 1998.

[Zukowski et al., 1993] M. Zukowski, A. Zeilinger, M.A. Horne, and A. Ek-

ert. ”event-ready-detectors” Bell experiment via entanglement swap-

ping. Phys. Rev. Lett. 71, 4287-4290, 1993.

[Zukowski et al., 1995] M. Zukowski, A. Zeilinger, and H. Weinfurter. Entan-

gling photons radiated by independent pulsed source. Ann. NY. Acad.

Sci. 755, 91-102, 1995.

BIBLIOGRAPHY 103

[Zukowski and Kaszlikowski, 1997] M. Zukowski, and D. Kaszlikowski. Crit-

ical visibility for N -particle Greenberger-Horne-Zeilinger correlations

to violate local realism. Phys. Rev. A 56, R1682-R1685, 1997.

[Zukowski, 1999] M. Zukowski. Violations of local realism in multiphoton

interference experiments. Phys. Rev. A, in press.

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