Entanglement-enhanced communication overa correlated-noise channel

Entanglement-enhanced communication overa correlated-noise channel

Andrzej DraganWojciech Wasilewski

Czesław RadzewiczWarsaw University

Jonathan BallUniversity of Oxford

Konrad BanaszekNicolaus Copernicus University Toruń, Poland

Alex LvovskyUniversity of Calgary

Squeezing eigenmodesin parametric down-conversion

Squeezing eigenmodesin parametric down-conversion

National Laboratory for Atomic, Molecular, and Optical Physics, Toruń, Poland

All that jazzAll that jazz

Sen

der

Receiv

er

Mutual information:

Channel capacity:

Depolarization in an optical fibreDepolarization in an optical fibre

Photon in a polarization state

H

V

H

V

1/2

1/2

1/2

1/2

2

1

Independently of the averaged output state has the form:

Capacity of coding in the polarization state of a single photon:

Random polarization transformation

Information codingInformation coding

H

H

V

VSender:

Probabilities of measurement outcomes:

H&H, V&V

H&V, V&H

2/3

2/3

1/3

1/3Capacity per photon pair:

Collective detectionCollective detection

Probabilities of measurement outcomes:

2&0, 0&2

1&1

1

1/2

1/2

Capacity:

Entangled states are useful!Entangled states are useful!

Probabilities of measurement outcomes:

2&0, 0&2

1&1

1

1

Capacity:

Proof-of-principle experimentProof-of-principle experiment

2&0, 0&2

1&1

1

1

Entangled ensemble:

2&0, 0&2

1&1

1

1/2

1/2

Separable ensemble:

These are optimal ensembles for separable and entangled inputs (assuming collective detection), which follows from optimizing the Holevo bound.

J. Ball, A. Dragan, and K.Banaszek,Phys. Rev. A 69, 042324 (2004)

Source of polarization-entangled pairsSource of polarization-entangled pairsP. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, Phys. Rev. A 60, R773 (1999)

For a suitable polarization of the pump pulses, the generated two-photon state has the form:

With a half-wave plate in one arm it can be transformed into:

or

Experimental setupExperimental setup

K. Banaszek, A. Dragan, W. Wasilewski, and C. Radzewicz, Phys. Rev. Lett. 92, 257901 (2004)

Triplet events:

D1 & D2 D3 & D4

Singlet events:

D1 & D3 D2 & D3 D2 & D3 D2 & D4

Experimental resultsExperimental results

Dealing with collective depolarization

Dealing with collective depolarization

1) Phase encoding in time bins: fixed input polarization, polarization-independent receiver.

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, Phys. Rev. Lett. 82, 2594 (1999).

2) Decoherence-free subspacesfor a train of single photons.

J.-C. Boileau, D. Gottesman, R. Laamme,D. Poulin, and R. W. Spekkens, Phys. Rev. Lett. 92, 017901 (2004).

General scenarioGeneral scenario

Physical system:

• arbitrarily many photons

• N time bins that encompass two orthogonal polarizations

• How many distinguishable states can we send via the channel?

• What is the biggest decoherence-free subspace?

Mathematical modelMathematical model

General transformation:

where:

– the entire quantum state of light across N time bins– element of U(2) describing the transformation of the polarization modes in a single time bin.– unitary representation of in a single time bin

We will decompose with

and

Schwinger representationSchwinger representation

...

Ordering Fock states in a single time bin according to the combined number of photons l:

Here is (2j+1)-dimensional representation of SU(2). Consequently has the explicit decomposition in the form:

Representation of

...

DecompositionDecomposition

Decomposition into irreducible representations:

Integration over removes coherence between subspaces with different total photon number L. Also, no coherence is left between subspaces with different j.

tells us:

• how many orthogonal states can be sent in the subspace j

• dimensionality of the decoherence-free subsystem

Recursion formula for :

J. L. Ball and K. Banaszek,quant-ph/0410077;Open Syst. Inf. Dyn. 12, 121 (2005)

Biggest decoherence-free subsystems have usually hybrid character!

QuestionsQuestions

• Relationship to quantum reference frames for spin systems [S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Phys. Rev. Lett. 91, 027901 (2003)]

• Partial correlations?

• Linear optical implementations?

• How much entanglement is needed for implementing decoherence-free subsystems?

• Shared phase reference?

• Self-referencing schemes? [Z. D. Walton, A. F. Abouraddy, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, Phys. Rev. Lett. 91, 087901 (2003)]

• Other decoherence mechanisms, e.g. polarization mode dispersion?

Multimode squeezingMultimode squeezing

–

Single-mode model:

SHG

PDC

Brutal reality (still simplified):

[See for example: M. Matuszewski, W. Wasilewski, M. Trippenbach, and Y. B. Band,Opt. Comm. 221, 337 (2003)]

Perturbative regimePerturbative regime

Schmidt decomposition for a symmetric two-photon wave function:

C. K. Law, I. A. Walmsley, and J. H. Eberly,Phys. Rev. Lett. 84, 5304 (2000)

We can now define eigenmodes which yields:

The spectral amplitudes characterize pure squeezing modes

The wave function up to the two-photon term:

W. P. Grice and I. A. Walmsley, Phys. Rev. A 56, 1627 (1997);T. E. Keller and M. H. Rubin, Phys. Rev A 56, 1534 (1997)

Decomposition for arbitrary pump

Decomposition for arbitrary pump

As the commutation relations for the output field operators must be preserved, the two integral kernels can be decomposed using the Bloch-Messiah theorem:

S. L. Braunstein, quant-ph/9904002;see also R. S. Bennink and R. W. Boyd,Phys. Rev. A 66, 053815 (2002)

Squeezing modesSqueezing modes

The Bloch-Messiah theorem allows us to introduce eigenmodes for input and output fields:

which evolve according to

• describe modes that are described by pure squeezed states

• tell us what modes need to be seeded to retain puritySome properties:

• For low pump powers, usually a large number of modes becomes squeezed with similar squeezing parameters

• Any superposition of these modes (with right phases!) will exhibit squeezing

• The shape of the modes changes with the increasing pump intensity! This and much more in a poster by Wojtek

Wasilewski

The EndThe End

TheoryTheory

Everything that emerges are Werner states

One-dimensional optimization problem for the Holevo bound

What about phase encoding?

Recursion formulaRecursion formula

Decompostion of the corresponding su(2) algebra:

If we subtract one time bin:

N bins, L photons

N-1 bins, L′ photons

......

L–L′photons

Direct sumDirect sum

The product of two angular momentum algebras has the standard decomposition as:

Therefore the algebra for L photons in N time bins can be written as a triple direct sum:

Decoherence-free subsystems

Decoherence-free subsystems

Rearranging the summation order finally yields:

Underlined entries with correspond to pure phase encoding (with all the input pulses having identical polarizations)– in most cases we can do better than that!