Universita degli Studi di Perugia

DIPARTIMENTO DI FISICA E GEOLOGIA

Corso di Laurea Triennale in Fisica

Tesi di Laurea Triennale

Entanglement Entropy

Candidato:

Federico GrasselliMatricola 256291

Relatore:

Prof. Gianluca Grignani

Anno Accademico 2013/2014

Sessione di Laurea 30 Settembre 2014

Contents

1 Density Operator 31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Pure States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Mixed States . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 Time Evolution of a mixed state . . . . . . . . . . . . . . . . 101.7 Density Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 111.8 Reduced Density Operator . . . . . . . . . . . . . . . . . . . . 13

2 The Density Operator Picture 152.1 The Measurement Postulate of Q.M. . . . . . . . . . . . . . . 152.2 Time Evolution and Measurements for Density Operators . . 172.3 New Q.M. Postulates . . . . . . . . . . . . . . . . . . . . . . . 18

3 Entanglement and Entanglement Entropy 203.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 Pure States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.1 Bipartite systems . . . . . . . . . . . . . . . . . . . . . 213.2.2 Multipartite systems . . . . . . . . . . . . . . . . . . . 32

3.3 Mixed States . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3.1 Bipartite systems . . . . . . . . . . . . . . . . . . . . . 343.3.2 Multipartite systems . . . . . . . . . . . . . . . . . . . 41

3.4 An interesting example . . . . . . . . . . . . . . . . . . . . . . 423.5 Monogamy and Area Law . . . . . . . . . . . . . . . . . . . . 44

4 Conclusions 46

Appendices 48

A The generic spin-12 state 49

B Density Matrix in the Position operator eigenbasis 51

1

C Shannon and von Neumann Entropy 52

Bibliography 54

2

Chapter 1

Density Operator

1.1 Introduction

Let’s consider a quantum system: we usually identify it through its state, i.e.its wave function. In this way, we assume it is possible to define the systemcompletely and build the function which represents it. However, this is notalways feasible. For instance, in an electron-target scattering experiment weuse the electron wave function to compute the cross section of the process;while if we want to estimate it experimentally, an electron beam –madeof many identical electrons prepared in the same initial state– is needed.Clearly, the experimental apparatus won’t produce electrons in the exactsame conditions, hence considering a single electron as representative of thewhole beam implies a process of idealization. Although such a process isacceptable in the majority of the cases, sometimes it is not: suppose thebeam was prepared without taking into account electron spin, therefore ameasure of Sz over all electrons will statistically result in 50% h/2 and50% −h/2 (neglecting the presence of polarizers). Consequently, we cannotspecify the spin wave function more than:

ψ =1√2

(χ+ + eiαχ−

)where α is unknown. A spin function like the one just written, or in generala function like

ψ = aχ+ + bχ−

(where a and b are fixed complex numbers satisfying |a|2+|b|2 = 1), describe1

an electron spin oriented in a particular direction n = n (θ, ϕ) with θ =2 arccos |a|, ϕ = arg b− arg a. This means that ψ can represent an electronbeam polarized along n while it cannot represent an unpolarized beamsuch as the one previously hypothesized. Moreover, since the latter is an

1proof in appendix A

3

incoherent superposition of states, it cannot be represented by any singlestate of which it is composed, contradicting the assumption made at thebeginning.

In these cases, that is when the lack of information about a collectionof single states prevents us from describing them one by one completely, wecan still describe such collection through a statistical analysis, in particularby means of the density operator [1].

1.2 Nomenclature

• Given a quantum system, if it can be identified by a complete set ofcommuting observables, its state is called a pure state. We call purestates even those states which are a coherent superposition – thuscompletely determined– of such states. In other words, every statewhich is represented by a definite wave function is a pure state;

• A collection of systems all described by the same pure state is namedpure ensemble;

• A mixture of different pure ensembles is named mixed ensemble, forexample the unpolarized beam.

Now we will go through the description of pure and mixed states madeby the density operator [1].

1.3 Pure States

Let |s〉 be a pure state representing a pure ensemble, normalized to unity. Wealready know how to compute expectation values and probabilities regardingsingle measurements of an observable F . Given the eigenvalue equation forF :

F |ϕn〉 = fn|ϕn〉 (1.1)

(assuming discrete eigenvalues for simplicity), |s〉 can be expressed as super-position of the eigenvectors of F :

|s〉 =∑n

cn|ϕn〉 (1.2)

wherecn = 〈ϕn|s〉. (1.3)

The probability of obtaining fn as a result of the measure of F on the systemin its state |s〉 is:

P(fn) = |cn|2 (1.4)

4

which may also be written, introducing the projector Pn of the nth eigen-vector: Pn = |ϕn〉〈ϕn| , as:

P(fn) = 〈s|ϕn〉〈ϕn|s〉 = 〈s|Pn|s〉 = 〈Pn〉s. (1.5)

The expectation value of F on the pure state |s〉 is:

〈F 〉s = 〈s|F |s〉 =∑n

|cn|2fn =∑n

〈ϕn|s〉〈s|ϕn〉fn =∑n

〈ϕn|s〉〈s|F |ϕn〉

(1.6)thus defining the operator:

ρ = |s〉〈s| (1.7)

it can be written:〈F 〉s =

∑n

〈ϕn|ρF |ϕn〉. (1.8)

The operator (1.7) is the density operator of a pure ensemble represented bythe pure state |s〉. Note that while a state, although normalized, is defined upto an arbitrary multiplicative factor, density operators are uniquely defined,thanks to their definition.

The (1.8) expression shows that the expectation value for F can beobtained computing the trace of ρF matrix:

〈F 〉s = Tr(ρF ) = Tr(Fρ) (1.9)

and since trace is invariant with respect to a change of basis2, the (1.9)has general validity. As a matter of fact, computing the trace of ρF in anarbitrary orthonormal basis {|αk〉} we obtain 〈F 〉s :

Tr(ρF ) =∑k

〈αk|s〉〈s|F |αk〉 =∑k

〈s|F |αk〉〈αk|s〉 = 〈s|F |s〉 = 〈F 〉s (1.10)

(where we used the completeness relation for {|αk〉}).Thanks to (1.9) we can compute P(fn) in the same way: (choosing

F = Pn)P(fn) = 〈Pn〉s = Tr(ρPn) = Tr(Pnρ) (1.11)

1.4 Mixed States

Let’s consider a mixed ensemble, that is, a set of pure ensembles, repre-sented by their pure state |s(i)〉, which constitute the former with differentpositive weights pi. Equivalently, we can consider a single quantum systemof the ensemble, which, therefore is in one of a number of states |s(i)〉 with

2hence trace is an intrinsic property of an operator, thus we won’t indicate matrixsymbols under trace sign anymore.

5

respective probabilities pi (for i = 1, ..., n). The “state” of such a quantumsystem is called a mixed state.

Note that the only assumption on the states |s(i)〉 is the normalizationto one, with no orthogonality required; obviously the statistical weights pisatisfy: 0 ≤ pi ≤ 1 and

∑i pi = 1.

The definition of the mean value of an observable F over a mixed stateis quite obvious:

〈F 〉 =∑i

pi〈F 〉i (1.12)

where 〈F 〉i = 〈s(i)|F |s(i)〉.Along the same lines, the probability that a measure of F over a mixed

state provides eigenvalue fn is:

P(fn) =∑i

piP(i)(fn) (1.13)

where P(i)(fn) = |cn(i)|2 = |〈ϕn|s(i)〉|2 and |s(i)〉 =∑

n cn(i)|ϕn〉.

As done in the previous section, we will now define the density operatorof a mixed state by making explicit the expectation value of F in the state|s(i)〉:

〈F 〉i = 〈s(i)|F |s(i)〉 =∑n

fn|〈ϕn|s(i)〉|2 =∑n

fn〈ϕn|s(i)〉〈s(i)|ϕn〉 (1.14)

which substituted in 〈F 〉:

〈F 〉 =∑i

pi∑n

fn〈ϕn|s(i)〉〈s(i)|ϕn〉 =∑n

〈ϕn|∑i

pi|s(i)〉〈s(i)|F |ϕn〉 (1.15)

permits us to define the density operator of a mixed state as follows:

ρ =∑i

pi|s(i)〉〈s(i)| (1.16)

and permits us to achieve the analogous expression of (1.9) in the mixedstate case:

〈F 〉 =∑n

〈ϕn|ρF |ϕn〉 = Tr(ρF ) = Tr(Fρ) [ρ is the one defined in (1.16)] (1.17)

We point out that:

• As one can see in expression (1.17) all information about the mixtureis factorized and contained in ρ;

• As for the density operator of a pure state, the density operator of amixed state is defined in a unique way, since arbitrary phase factorswould vanish if factorized out of the projectors |s(i)〉〈s(i)|.

6

By means of the density operator one can also compute the probabilityP(fn) as in (1.11):

P(fn) =∑i

piP(i)(fn)

=∑i

pi|〈ϕn|s(i)〉|2

=∑i

pi〈s(i)|ϕn〉〈ϕn|s(i)〉

=∑i

pi〈s(i)|Pn|s(i)〉

=∑i

pi〈Pn〉i

= 〈Pn〉 [choosing F = Pn]

= Tr(ρPn) = Tr(Pnρ) (1.18)

1.5 Properties

Now let’s list some properties of density operators. Each property is satis-fied both by density operators of pure and mixed states, unless otherwisespecified.

1. ρ = ρ† : follows from the definition in (1.16) since pi ∈ < ∀i ;

2. Tr(ρ) = 1 :

Tr(ρ) =∑k

〈αk|∑i

pi|s(i)〉〈s(i)|αk〉 =∑i

pi∑k

|〈s(i)|αk〉|2

=

(Parseval’s equation assures us that∑

k |〈s(i)|αk〉|2

= 〈s(i)|s(i)〉 = 1)

=∑i

pi = 1;

3. ρ is positive-semidefinite: given an arbitrary vector |a〉, 〈a|ρ|a〉 =∑i pi〈a|s(i)〉〈s(i)|a〉 =

∑i pi|〈a|s(i)〉|2 ≥ 0;

4. eigenvalues satisfy 0 ≤ λk ≤ 1 ∀k : since ρ is self-adjoint, one can al-ways diagonalize ρ and notice that its diagonal elements are its eigen-values; now, using property 3 one has that every diagonal element, i.e.every eigenvalue, is non-negative. The eigenvalue equation for ρ is:ρ|λk〉 = λk|λk〉. If we compute Tr(ρ) in the eigenvector basis {|λk〉}we obtain the sum of its eigenvalues:

∑k λk, and thanks to property

2 we have: ∑k

λk = 1. (1.19)

7

In conclusion we have seen that: 0 ≤ λk ≤ 1 ∀k.N.B.: if the eigenvalue λk is equal to 1, then (1.19) shows that allthe others are 0; this is the particular case of a pure state, as amatter of fact, writing ρ through its spectral representation: ρ =∑

k λk|λk〉〈λk| = |λk〉〈λk| which is the definition of pure state.

5. Tr(ρ2) ≤ 1 : spectral representation for ρ2 is:

ρ2 =∑k

λk2|λk〉〈λk|

which in general differs from ρ, computing Tr(ρ2) =∑

k λk2 and con-

sidering property 4 and (1.19), one has:

Tr(ρ2) ≤ 1,

where equality occurs for pure states: recalling previous N.B., all eigen-values are zero except for one which equals 1, hence

ρ2 = λk2|λk〉〈λk| = |λk〉〈λk| = ρ,

now thanks to property 2, one has:

Tr(ρ2) = Tr(ρ) = 1.

Therefore relations

ρ2 = ρ and Tr(ρ2) = 1

are both sufficient to determine whether a state is pure.

The following example [2, page 15], in which we will verify property 5,shows the fundamental difference between coherent superposition of quan-tum states –realized through state-vector addition– and the mixing of them–realized through statistical operator addition–:

• A coherent superposition of states leads to a pure state:

| ↗〉 =1√2

(| ↑〉+ | ↓〉)

and this can be easily verified by computing Tr(P 2(| ↗〉)) =Tr(P (| ↗〉)) = 1.

• Considering instead the normalized sum of the two projectors P (| ↑〉)and P (| ↓〉):

ρ =1

2(P (| ↑〉) + P (| ↓〉))

we obtain a mixed state, confirmed by computing Tr(ρ2) =Tr(

14

[P 2(| ↑〉) + P 2(| ↓〉)

])= 1

4Tr(P (| ↑〉) + P (| ↓〉)) = 12 < 1.

8

Such a difference may be better visualized through an ideal experiment:consider many identically prepared systems, such as electrons, directed nor-mally toward a Young double-slit interferometer, and, if not absorbed, sub-sequently detected on an opaque screen. Let a1(x) be the probability am-plitude –i.e. the wave function– corresponding to the passage through thefirst slit toward the spatial point x on the measurement screen, which, likethe interferometer, is perpendicular to the direction of the initial beam.Similarly, let a2(x) be the probability amplitude corresponding to passagethrough the second slit toward the spatial point x. The amplitude for sys-tems being found at x when both slits are passable, so that either slitmay be entered on the way to the screen, is:

a12(x) =1√2

(a1(x) + a2(x)) (1.20)

according to the superposition principle (the factor 1/√

2 confers propernormalization on the total probability). The probability density of findingthe quantum systems at point x on the collection screen upon measurementis the complex square of the amplitude:

p12(x) = |a12(x)|2 =1

2

[|a1(x)|2 + |a2(x)|2 + 2Re(a1(x)∗a2(x))

](1.21)

and refers to the coherent superposition of quantum states a1(x) and a2(x),which clearly interfere.

Suppose now that the experimental apparatus is modified so that onlyone slit is available at a time, with both being made available in thefull course of the experiment on the entire ensemble of systems. Assumingthat diffraction from each slit is negligible, interference previously detectedvanishes. As a matter of fact in this case there is no quantum superpositionof states, but rather a mixing of them: the modified apparatus implies thata certain population p of quantum systems –arbitrarily determined by theexperimentalist– is in state a1(x), while the remaining population 1−p is instate a2(x). The probability density of finding one of the quantum systemsat point x on the collection screen upon measurement is3:

ρxx =∑i

pi|ai(x)|2 (1.22)

which yields, taking p = 1− p = 1/2:

ρxx =1

2

[|a1(x)|2 + |a2(x)|2

]. (1.23)

Comparing (1.21) with (1.23), the striking difference between quantum su-perposition and quantum mixing shows up: the quantum interference char-acterizing the former is missing in the latter.

3proof in appendix B

9

We would like to point out one last important fact regarding densitymatrices. When talking about statistical ensembles of quantum systems,we necessarily deal with two distinct kinds of probability: statistical prob-abilities pi which appear explicitly in ρ’s definition, and those (P(i)(fn))which characterize the expectation value of observables measured on eachpure subsystem of the mixture (〈F 〉i =

∑n fnP(i)(fn)) and are implicit in

the vectors |s(i)〉. Consequently, when we compute traces in the density op-erator formalism, we refer to particular representations of it, dealing withdensity matrices depending on both kinds of probabilities indistinguishably.As a result, while one can uniquely define a density operator starting froma given mixture of states, one cannot associate a definite mixture of sys-tems to a given density matrix; therefore there can be different systemensembles described by the same density matrix [1].

1.6 Time Evolution of a mixed state

Given a mixed state described by its density operator:

ρ =∑i

pi|s(i)〉〈s(i)| (1.24)

in the Schrodinger picture we conclude ρ is time-dependent. Indeed, assum-ing our ensemble isolated (not interacting with the outside world) statisticalweights pi are time-independent, while states |s(i)〉 evolve over time accord-ing to:

d

dt|s(i)(t)〉 =

1

ihH|s(i)(t)〉. (1.25)

Deriving (1.24) and using (1.25):

dρ

dt=∑i

pi

(1

ihH|s(i)(t)〉〈s(i)(t)| − 1

ih|s(i)(t)〉〈s(i)(t)|H

)(1.26)

which yields:dρ

dt=

1

ih[H, ρ]. (1.27)

This last equation is called the von Neumann equation or Liouville-von Neu-mann equation and describes how a density operator evolves in time. Thisequation is the quantum analogous of the Liouville equation valid in classicalstatistical mechanics:

dρcldt

= {H, ρcl},

where ρcl(p, q) is the phase space distribution of the system. Note that,although identical up to a minus sign, the von Neumann equation and timeevolution of a Heisenberg operator A:

dA

dt=

1

ih[A,H]

10

are completely disconnected: while the former was obtained in the Schrodin-ger picture, the latter arises in a context where operators are, in general,time-dependent, whereas states are not; this means that in the Heisenbergpicture the density operator –as defined in (1.24)– is time-independent aswell [1].

At last, one can wonder whether (1.27) and the Ehrenfest theorem arereconcilable, since both belong to the Schrodinger picture. One can imme-diately verify they are self-consistent by deducing one through the other:

d

dt〈A〉 = [using (1.17)]

d

dtTr(ρA)

= Tr

(dρ

dtA

)=

1

ihTr([H, ρ]A)

=1

ihTr(HρA− ρHA) =

1

ihTr(ρAH − ρHA)

=1

ihTr(ρ[A,H]) =

1

ih〈[ρ,H]〉

which is the Ehrenfest theorem.In case of a time-independent Hamiltonian, the integrated form of the

von Neumann equation is:

ρ(t) = e−i(H/h)tρ(0)ei(H/h)t (1.28)

and it can be easily proved by recalling time evolution of a state for time-independent Hamiltonians:

|s(i)(t)〉 = e−i(H/h)t|s(i)(0)〉

and inserting this expression in (1.24):

ρ(t) =∑i

pi|s(i)(t)〉〈s(i)(t)|

=∑i

pie−i(H/h)t|s(i)(0)〉〈s(i)(0)|ei(H/h)t

= e−i(H/h)tρ(0)ei(H/h)t.

1.7 Density Matrices

In this section we will investigate the physical meaning of the matrix ele-ments of a density operator and then furnish a couple of notable examplesof density matrices [1].

Given an arbitrary orthonormal basis {|αk〉}, the generic matrix elementof ρ is:

ρkl = 〈αk|ρ|αl〉 =∑i

pi〈αk|s(i)〉〈s(i)|αl〉,

11

noting that

|s(i)〉 =∑k

ck(i)|αk〉 with ck

(i) = 〈αk|s(i)〉

we can rewrite it:ρkl =

∑i

pick(i)cl

(i)∗. (1.29)

• The diagonal elements are ρkk =∑

i pi|ck(i)|2; each ρkk is given by thesum of products between probability pi that a generic system belonging

to the mixture is in state |s(i)〉 and probability |ck(i)|2 of finding thestate |αk〉 in the state |s(i)〉. Hence, ρkk is the global probability offinding a generic element of the mixture in the state |αk〉. For thisreason, diagonal elements are called relative populations of the states|αk〉. Obviously

∑k ρkk = 1 due to property 2 of density operators.

• Regarding off-diagonal matrix elements, coefficients ck(i)cl

(i)∗ expressinterference effects between the states |αk〉 and |αl〉 which can arise ifthe pure state |s(i)〉 is a linear coherent superposition of such states.Therefore, the element ρkl represents the average of these interferenceeffects over the mixed ensemble, and can result in zero even if singu-larly each coefficient ck

(i)cl(i)∗ is not. In a statistical mixture of states

the averaging can cancel out interference effects. Vice versa, ρkl 6= 0means that even at the mixture level there are residual effects of co-herence between |αk〉 and |αl〉. For this reason, off-diagonal matrixelements are called coherences.

Clearly, both populations and coherences depend on the chosen basis {|αk〉}.Coherences are bounded from above by populations in the following way:

|ρkl|2 ≤ ρkkρll (1.30)

A remarkable case is when, having a time-independent Hamiltonian,{|αk〉} are its eigenvectors. In this case the population of the state |αk〉expresses the probability of finding a mixture element with energy Ek. Sucha probability is also time-independent; indeed, using (1.28) one has:

ρkk(t) = ρkk(0) (1.31)

ρkl(t) = eih

(El−Ek)tρkl(0) (1.32)

where we also notice that coherences oscillate in time with Bohr frequencies.

It is particularly interesting to analyze the matrix representation of twoopposite density operators: a completely mixed state and a pure state.

12

1. In a completely mixed state, all states must result equally populatedand there are no coherences. By naming the dimension of H as N({|αk〉} ∈ H), it follows that the population of each state is 1/N ,hence ρ is the identity matrix multiplied by 1/N (ρkk = 1/N andρkl = 0 for l 6= k);

2. Concerning a pure state identified by |s〉, let’s represent its densitymatrix in the basis {|αk〉}. Expressing |s〉 as:

|s〉 =∑k

ak|αk〉

it follows that:ρ =

∑k,l

akal∗|αk〉〈αl|

which has matrix elements:

ρkl = akal∗. (1.33)

So, in general, density matrix of a pure state has populations andcoherences different from zero. However, if one of the basis vectorsis |s〉, let’s say |s〉 ≡ |αk〉, then ak = δkk, al = δlk, which insertedin (1.33) return a matrix with all zero elements except for the k-thdiagonal element, that equals 1.

1.8 Reduced Density Operator

Given a mixed state C represented by density operator ρ, suppose we want tomeasure the observable FA which operates only on A subset of C, and let Bbe the complementary set of A in C. What can we say about the expectationvalue of FA? In order to answer, we first define the following orthonormalbases: {|αk〉} ∈ A, {|βl〉} ∈ B, {|γkl〉} ∈ C where |γkl〉 ≡ |αk〉|βl〉. Nextwe define an extension of FA which operates in the entire space of C: F =FA⊗IB. Then we define the expectation value of FA on C as the expectationvalue of F on C:

〈FA〉 ≡ 〈F 〉 = Tr(ρF ) =∑kl

〈γkl|ρF |γkl〉 = (completeness of {|γkl〉})

=∑kl

∑jm

〈γkl|ρ|γjm〉〈γjm|FA ⊗ IB|γkl〉 = (|γkl〉 ≡ |αk〉|βl〉)

=∑kljm

〈αk|〈βl|ρ|βm〉|αj〉〈βm|βl〉〈αj |FA|αk〉 = (sum over m)

=∑kj

〈αk|∑l

〈βl|ρ|βl〉|αj〉〈αj |FA|αk〉.

13

Denoting ρA the partial trace computed in the subspace containing subsetB, i.e.:

ρA = TrBρ =∑l

〈βl|ρ|βl〉 (1.34)

we have:

〈FA〉 =∑kj

〈αk|ρA|αj〉〈αj |FA|αk〉 (completeness of {|αj〉})

=∑k

〈αk|ρAFA|αk〉 = Tr(ρAFA) (1.35)

As one can see, the expectation value of FA has been manipulated achievingan analogous form of (1.17) by means of the operator ρA, named reduceddensity operator. The reduced density operator works only on the subspacecontaining subset A and satisfies all the properties already listed for a genericdensity operator [1].

14

Chapter 2

The Density OperatorPicture

In this chapter we will briefly show how all the postulates of Q.M. givenin the state vector language can be reformulated in terms of the densityoperator language, since some of them will be useful afterwards. The twoformulations are equivalent and both arise in the context of Schrodinger’spicture, however it is sometimes much easier to approach problems from onepoint of view rather than the other [3].

2.1 The Measurement Postulate of Q.M.

Let’s start by giving a slightly different version of the Measurement postulateof Q.M. than the one usually encountered in a first study of the state vectorlanguage. Note that it still refers to the state vector picture!

One knows that closed quantum systems evolve according to unitary evo-lution; nevertheless, when an experimentalist observes the system to carryout a measure of some physical magnitude, the interaction which followsmakes the system no longer closed, thus not subjected to unitary evolution.The following postulate provides a means for describing the effects of mea-surements on quantum systems: (number 3 refers to the numeration we willgive at the end of the chapter)

Postulate 3: Quantum measurements are described by a collection {Mm} of mea-surement operators acting on the state space of the system being mea-sured. The index m refers to the measurement outcomes that mayoccur in the experiment. If the state of the quantum system is |ψ〉immediately before the measurement, then the probability that resultm occurs is given by:

p(m) = 〈ψ|Mm†Mm|ψ〉 (2.1)

15

and the state of the system after the measurement is:

|ψ′〉 =Mm|ψ〉√p(m)

(2.2)

Furthermore the measurement operators satisfy the completeness equa-tion: ∑

m

Mm†Mm = I, (2.3)

such an equation expresses the fact that probabilities sum to one:∑m

p(m) =∑m

〈ψ|Mm†Mm|ψ〉 = 〈ψ|ψ〉 = 1

(assuming normalized state representing our system).

Obviously this postulate cannot be in contradiction with more widely-knownassumptions of Q.M.; as a matter of fact, we can obtain them in the partic-ular case of projective measurements.

Projective m.ts: A projective measurement is described by an observable M , a Hermi-tian operator on the state space of the system being observed. Theobservable has a spectral decomposition:

M =∑m

mPm

where Pm = |m〉〈m| is the projector onto the eigenspace of M witheigenvalue m. These kind of measurements can be understood as aspecial case of Postulate 3, that is when Mm = Pm.

Inserting Mm = Pm in (2.1):

p(m) = 〈ψ|Mm†Mm|ψ〉 = (Mmψ,Mmψ) = ‖Mmψ‖2 = ‖|m〉〈m|ψ〉‖2

= (‖|m〉‖2 = 1 and 〈m|ψ〉 = c(m), being |ψ〉 =∑

m c(m)|m〉)= |c(m)|2

which is the well-known probability for outcome m.Inserting Mm = Pm in (2.2):

|ψ′〉 =Mm|ψ〉√p(m)

=|m〉〈m|ψ〉|〈m|ψ〉|

= ±|m〉 (2.4)

which is, up to a sign, the known final state after a measurement.

16

2.2 Time Evolution and Measurements for Den-sity Operators

What we are going to show is that the postulates of Q.M. related to unitaryevolution and measurement can be rephrased in the language of densityoperators. In the next section we complete this rephrasing by giving anintrinsic characterization of the density operator that doesn’t rely on theidea of a state vector.

Time evolution can be easily described in the density operator language:suppose that the evolution of a closed quantum system is determined bythe unitary operator U . If the system was initially in the state |ψi〉 withprobability pi, then after the evolution has occurred the system will be inthe state U |ψi〉 with probability pi. Therefore the evolution of the densityoperator is described by the equation:

ρ =∑i

pi|ψi〉〈ψi|U−→∑i

piU |ψi〉〈ψi|U † = UρU † (2.5)

Measurements are also readily described in this context: suppose we performa measurement defined by measurement operators Mm. If the initial statewas |ψi〉, then the probability of getting result m is (2.1):

p(m|i) = 〈ψi|Mm†Mm|ψi〉 =

∑j

cj∗〈uj |Mm

†Mm|ψi〉 = (cj∗ = 〈ψi|uj〉)

=∑j

〈uj |Mm†Mm|ψi〉〈ψi|uj〉

= Tr(Mm†Mm|ψi〉〈ψi|) (2.6)

hence the total probability of getting m in any of the subsystems is:

p(m) =∑i

p(m|i)pi

=∑i

piTr(Mm†Mm|ψi〉〈ψi|)

= Tr(Mm†Mmρ)(which is a generalization of (1.18)) (2.7)

What is the density operator of the system after obtaining the measurementresult m? If the initial state was |ψi〉 then the state after getting m is (2.2):

|ψim〉 =Mm|ψi〉√p(m|i)

(2.8)

Thus, after a measurement which yields the result m, we have an ensemble ofstates |ψim〉 with respective probabilities p(i|m). The corresponding density

17

operator ρm is:

ρm =∑i

p(i|m)|ψim〉〈ψim| =

=∑i

p(i|m)Mm|ψi〉〈ψi|Mm

†

p(m|i)=

(by the elementary probability theory: p(m, i) = p(i|m)p(m) = p(m|i)p(i) ,so: p(i|m) = p(m, i)/p(m) = p(m|i)pi/p(m))

=∑i

piMm|ψi〉〈ψi|Mm

†

p(m)=

=MmρMm

†

Tr(Mm†Mmρ)

(2.9)

Before moving to the next section we want to remark that stating thata quantum system is prepared in the state ρi with probability pi means it isdescribed by the density operator ρ =

∑i piρi. Indeed, although prepared

in ρi with probability pi, there is a probability 1 − pi of finding it in manyother states ρi′ with respective probabilities pi′ .

2.3 New Q.M. Postulates

In the previous chapter we have seen that density operators defined by (1.16)satisfy properties 1,2,3 on page 7. Conversely, we could use these propertiesin order to define a density operator to be a Hermitian semidefinite-positiveoperator with trace equal to one. Making this definition allows us to refor-mulate the postulates of Q.M. in the density operator picture, unbinding itfrom the state vector idea.

Postulate 1 Associated to any isolated physical system is a Hilbert spaceknown as the state space of the system. The system is completely describedby its density operator, which is a Hermitian semidefinite-positive operatorwith trace equal to one, acting on the state space of the system. If a quantumsystem is in the state ρi with probability pi, then the density operator for thesystem is

∑i piρi.

Postulate 2 The evolution of a closed quantum system is described by theunitary transformation (2.5):

ρ′ = UρU † (2.10)

18

Postulate 3 (Generalized Measurements) Quantum measurements aredescribed by a collection {Mm} of measurement operators acting on the statespace of the system being measured. The index m refers to the measurementoutcomes that may occur in the experiment. If the state of the quantumsystem is ρ immediately before the measurement, then the probability thatresult m occurs is given by (2.7):

p(m) = Tr(Mm†Mmρ) (2.11)

and the state of the system after the measurement is (2.9):

ρm =MmρMm

†

Tr(Mm†Mmρ)

(2.12)

The measurement operators satisfy the completeness equation:∑m

Mm†Mm = I. (2.13)

Postulate 4 The state space of a composite physical system is the tensorproduct of the state spaces of the component physical systems. Moreover, ifwe have systems numbered 1 through n, and system number i is prepared inthe state ρi, then the joint state of the total system is ρ1 ⊗ ρ2 ⊗ ...⊗ ρn.

19

Chapter 3

Entanglement andEntanglement Entropy

3.1 Introduction

Entanglement is arguably the most fundamental – according to Schrodinger– and potentially disturbing characteristic distinguishing the quantum fromthe classical world. One could describe it as follows: it implies that themeasurement of an observable of a subsystem may affect drastically andinstantaneously the possible outcome of a measurement on another part ofthe system, no matter how far apart it is spatially. This is to be distinguishedfrom the phenomenon of classical correlation, where the distribution of thepossible outcomes of a measurement on one part of the system may dependon the outcome of a previous measurement elsewhere, but is strictly limitedby the signal propagation speed [7].

In order to better understand the above description, let’s immediatelypresent a concrete example of entanglement: the singlet state of two spin-1

2particles

|s0〉 =1√2

(| ↑〉1 ⊗ | ↓〉2 − | ↓〉1 ⊗ | ↑〉2) (3.1)

where subscripts 1,2 refer respectively to the first and second particle. If anobserver measures the spin of the first particle and obtains a particular value,say σz1 = +1/2, the spin value of the second particle is automatically fixedto σz2 = −1/2, since after the measurement the system has collapsed intoits eigenstate | ↑〉1 ⊗ | ↓〉2. This phenomenon implies a sort of non-classicalcorrelation between the two particles, i..e. entanglement. The correspondingstate is called entangled. Obviously entanglement is not always guaranteedin all two-spin systems, for instance in the triplet:

|t1〉 = | ↑〉1 ⊗ | ↑〉2 (3.2)

measuring the spin of the first particle doesn’t give additional information

20

about the state of the second particle, hence the two particles are not en-tangled, and their state is called separable.

Entanglement can be defined for both pure and mixed states. In the firstcase the definition of entanglement poses no difficulty and can be naturallyunderstood in terms of standard correlations. In the second, however, thedefinition is subtle, as correlations do not directly imply entanglement. Atthe same time, for two objects, entanglement can be easily defined and quan-tified, while for many objects this is not so simple: new correlations mayappear, which cannot be reduced to the case of two objects. We will intro-duce separately entanglement for pure and mixed states, considering in eachthe case of bipartite systems (i.e. two objects) and multipartite ones. Moredetails can be found in [6].

3.2 Pure States

We start out by considering the simplest case, namely the entanglementpresent in a many-body quantum pure state Ψ. This is the case, for instance,at zero temperature (if there’s no degeneracy), or in most applications inquantum information science. Unfortunately this is an idealized case, sincezero temperature cannot be reached in practice and also interaction with theenvironment causes a non-unitary evolution of the pure state into a mixedone, a process referred to as decoherence.

3.2.1 Bipartite systems

Given two systems A and B, we denote by HA and HB the correspondingHilbert spaces, and by {|n〉X} an orthonormal basis in HX , where n =1, 2, ...dX with dX = dim(HX ). Unless we state differently, we will alwayswork with spin-1

2 particles, where dA = dB = 2 (generalization to higherdimensions is straightforward). These systems are also called qubits sincethey are widely used in quantum information theory, so we will take asa basis the standard one: {|0〉X , |1〉X}. The Hilbert space correspondingto the whole system is H = HA ⊗ HB (according to Postulate 4). Anorthonormal basis in that space is {|n〉A ⊗ |m〉B} that for simplicity will bewritten {|n,m〉}. For instance, any pure state representing two subsystemsA and B can be written as:

|Ψ〉 =∑n,m

cn,m|n,m〉∑n,m

|cn.m|2 = 1.

We will consider observables for each of the subsystems, which will berepresented as operators acting on the corresponding spaces. For example,σA1 will stand for σA1 ⊗ IB, which denotes an operator σ1 acting on A and

21

I acting on B. Pauli operators acting on qubits will often appear; they aredefined as follows:

σx = |0〉〈1|+ |1〉〈0|σy = i(|0〉〈1| − |1〉〈0|)σz = |1〉〈1| − |0〉〈0| = −iσxσy

Definition 1 We say that Ψ ∈ H is a product state if there exist twovectors ϕ1 ∈ HA and ϕ2 ∈ HB such that |Ψ〉 = |ϕ1〉A ⊗ |ϕ2〉B. Otherwise,we say that Ψ is an entangled state.

This definition is a particular case of a more general one given by Schrodingerfor entangled pure states of multipartite systems, which we will see in thenext subsection. Examples of product states are those forming the orthonor-mal basis |n,m〉. Examples of entangled states are the so-called Bell states:

|Φ±〉 =1√2

(|0, 0〉 ± |1, 1〉) (3.3)

|Ψ±〉 =1√2

(|0, 1〉 ± |1, 0〉) (3.4)

Quantum correlations. The difference between entangled and productstates is that the first give rise to correlations. If we have a product state|Ψ〉 = |ϕ1, ϕ2〉 then the expectation value 〈Ψ|σA1 ⊗ σB2 |Ψ〉 factorizes into〈ϕ1|σ1|ϕ1〉〈ϕ2|σ2|ϕ2〉, therefore the result of measurements in both systemswill be uncorrelated. For an entangled state, on the contrary, there alwaysexist a couple of observables in A and B whose expectation value does notfactorize and, thus, for which the results of measurements will be correlated.For instance, let’s consider the Pauli operator along the direction ~n as:

σ~n = nxσx + nzσz (where ~n is a unity vector with ny = 0) (3.5)

One can see that:〈Φ+|σA~n ⊗ σ

B~m|Φ

+〉 = ~n · ~m (3.6)

whereas 〈Φ+|σA~n |Φ+〉 = 0. Thus, whenever we measure the same Pauli op-

erators in A and B (i.e. when ~n and ~m are parallel) the results are randombut maximally correlated; that is, we have the same outcome in A and B.This statement, however, was obvious if applied to a measure of σAz and σBzon Φ+, due to its definition (3.3). For this reason, Φ+ and the other Bellstates are usually referred to as maximally entangled states; they give riseto the maximal quantum correlations.

Schmidt decomposition (SD). In order to analyze entanglement in bi-partite systems, it is useful to introduce the Schmidt decomposition. Given

22

a generic pure state Φ representing two subsystems A and B it is alwayspossible to find an orthonormal basis {|un〉} in HA and {|vm〉} in HB suchthat:

|Φ〉 =∑

k≤dA,dB

dk|uk, vk〉 (3.7)

where elements dk are called Schmidt coefficients.A proof of this theorem ([4]) could be the following: any pure state Φrepresenting two subsystems A and B can be written as:

|Ψ〉 =∑n,m

cn,m|n〉A|m〉B =∑n

|n〉A|n〉B (3.8)

where {|n〉A} and {|m〉B} are orthonormal bases and having defined |n〉B =∑m cn,m|m〉B which are not necessarily orthonormal. Now suppose that

{|n〉A} is an eigenbasis1 of ρA (reduced density operator) with eigenvalues{dn2}. Then spectral representation for ρA is:

ρA =∑n

dn2|n〉A〈n|A (3.9)

By definition, we can compute ρA through the partial trace of ρ = |Φ〉〈Φ|on B:

ρA = TrB(|Φ〉〈Φ|) = (3.8) = TrB

(∑n,m

|n〉A〈m|A ⊗ |n〉B〈m|B

)=

∑n,m

|n〉A〈m|ATrB (|n〉B〈m|B) =∑n,m

|n〉A〈m|A∑k

(〈k|n〉〈m|k〉)B

=∑n,m

|n〉A〈m|A∑k

(〈m|k〉〈k|n〉)B = (completeness of {|k〉})

=∑n,m

|n〉A〈m|A〈m|n〉B (3.10)

Comparing (3.9) with (3.10) one deduces:

〈m|n〉B = dn2δmn (3.11)

which assures us that {|n〉B} is an orthogonal basis. We can normalize it byrescaling: |n′〉B = 1

dn|n〉B for all non-zero eigenvalues. From (3.8) we finally

have:|Ψ〉 =

∑n

|n〉A|n〉B =∑n

dn|n〉A|n′〉B (3.12)

that proves the theorem.Some notes ([4]):

1an orthonormal eigenbasis of ρA can always be found since ρA is a self-adjoint operator.Furthermore, since it is semidefinite positive, its eigenvalues are non-negative.

23

1. Every pure state can be decomposed according to SD, however whilethis decomposition: |Ψ〉 =

∑n,m cn,m|n,m〉 can be realized with any

orthonormal basis, the basis used in SD depends on the pure state tobe decomposed, as the proof highlights.

2. Computing the reduced density operator of the system B:

ρB = TrA(|Φ〉〈Φ|) = ... =∑k

|k〉B〈k|B =∑n

dn2|n′〉B〈n′|B (3.13)

we find out that ρB has the same non-zero eigenvalues of ρA althoughdimensions ofHA andHB might be different, in which case the numberof null eigenvalues will differ. In conclusion, The square of theSchmidt coefficients are the non-zero eigenvalues of both ρAand ρB.

3. Every pure state is associated with its Schmidt number, which is thenumber of non-zero eigenvalues of ρA, hence the number of terms com-posing the state’s SD. If the Schmidt number is equal to 1, then thestate is a product state, also called a separable state. If the Schmidtnumber is greater than 1, then the state is entangled, since it cannotbe written as a tensor product of two vectors.

4. It is important to stress the fact that if the pure state representing thewhole system is separable, then the reduced density matrices ρA andρB describe pure subsystems (because they are in the form (1.7));if the pure state is entangled, then the reduced density matricesdescribe mixed subsystems (because they are in the form (1.16)).

Consider the following example:

|Φ〉 = cos(θ)|0, 0〉+ sin(θ)|1, 1〉 (3.14)

Φ is already written in the the SD form. The eigenvalues of the reduced den-sity operator are cos(θ)2, sin(θ)2; for θ = 0 we have a product state, whereasfor θ = π/4 we have the Bell state Φ+ , which is a maximally entangledstate, as explained in the previous paragraph. In parallel, the reduced den-sity operators get more and more mixed as one increases θ from 0 to π/4(note that the purity of a mixed state ρ is related to the distribution if itseigenvalues when considered as probabilities).

Entanglement entropy. The previous example shows that entanglementof a pure state is linked to the mixedness of the reduced density operators.As a consequence, we could use this relation to introduce a measure of entan-glement of a pure state by using any measure of mixedness of its subsystems.

24

A natural way of measuring the latter2 is the von Neumann entropy:

S(ρ) = −Tr(ρ ln ρ) (3.15)

Thus, we define the entropy of entanglement of a pure state Φ as:

E(Φ) = S(ρA) = S(ρB) = −∑k

dk2 ln dk

2 (3.16)

For a product state E = 0 (since Schmidt number equals 1 and d =1), whereas the maximum entanglement is E = log[min(dA, dB)] which isreached for the state for which all dk are equal to 1/

√min(dA, dB). These

are thus called maximally entangled states. One can easily verify that theBell states are maximally entangled states, as already pointed out.

We want to emphasize the fact that, in order to evaluate the amountof entanglement of the total pure state Φ, we quantify the v.N. entropy ofits subsystems A and B, rather than the v.N. entropy of the whole state Φ,which is always zero for pure states. This because entanglement is strictlyrelated to the correlation between the two subsystems, which can be visual-ized through a measure of their v.N. entropy.

Finally, the fact that the v.N. entropy of a pure state is zero is quiteobvious, considering that such entropy has been defined to measure themixedness of a state. Furthermore, supposing that a quantum pure systemhas a pure subsystem, such a system is necessarily separable (as noticed inthe SD paragraph), hence E = 0, which means that the v.N. entropy of itspure subsystems is null. Alternatively, one could simply compute S(ρ) usingfor the pure state ρ the representation given in section 1.7.

Let’s compute the entropy of entanglement of the following parametricpure state ([3, page 504]):

|Φ〉 =√p| ↑〉 ⊗ | ↓〉+

√1− p| ↓〉 ⊗ | ↑〉 (3.17)

which is completely separable for p = 0, 1 and maximally entangled for p =1/2. The reduced density matrix related to the first subsystem can be easilyobtained, noticing that the vector basis |n〉A used for subsystem A in thegeneric SD (3.12) is the same basis which diagonalizes ρA in (3.9). Therefore,looking at (3.17) we can immediately write ρA’s spectral decomposition:

ρA = p| ↑〉〈↑ |+ (1− p)| ↓〉〈↓ |

We can then compute:

E(Φ) = S(ρA) = −p log(p)− (1− p) log(1− p) (3.18)

(we don’t specify the logarithm base voluntarily, it could be either 2 ore). Below we plotted E(Φ) as a function of p; note that the pure state is

2more info in Appendix C

25

maximally entangled for p = 1/2, as we expected.

Before giving a physical interpretation of the entropy of entanglement, wewill highlight some basic properties of the von Nuemann entropy [3].

Property 1 Suppose a composite system AB is in a pure state. ThenS(A) = S(B).

Property 2 Suppose pi are probabilities, and the states ρi have support onorthogonal subspaces. Then

S

(∑i

piρi

)= H(pi) +

∑i

piS(ρi).

Property 3 (Joint entropy theorem) Suppose pi are probabilities, |i〉are orthogonal states for a system A, and ρi is any set of density opera-tors for another system B. Then

S

(∑i

pi|i〉〈i| ⊗ ρi

)= H(pi) +

∑i

piS(ρi).

Property 4 (Entropy of a tensor product) Suppose ρ is a density op-erator acting on system A, while σ is a density operator acting on systemB. Then

S(ρ⊗ σ) = S(ρ) + S(σ).

26

Proof

1. From the SD we know that the eigenvalues of the density operators ofsystems A and B are the same. The entropy is determined completelyby these eigenvalues, so S(A) = S(B).

2. Let λji and |eji 〉 be the eigenvalues and corresponding eigenvectors of

ρi. Observe that piλji and |eji 〉 are the eigenvalues and eigenvectors of

piρi, and since different ρis belong to orthogonal subspaces, piλji and

|eji 〉 will also be the eigenvalues and eigenvectors of∑

i piρi. Thus

S

(∑i

piρi

)= −

∑ij

piλji log piλ

ji

= −∑i

pi log pi∑j

λji −∑i

pi∑j

λji log λji

(∑

j λji = 1, see (1.19))

= H(pi) +∑i

piS(ρi)

3. As in the previous proof, let piλji and |eji 〉B be the eigenvalues and

corresponding eigenvectors of piρi. Clearly, the normalized eigenvectorof the operator |i〉〈i| is |i〉A with eigenvalue 1. Hence piλ

ji and |i〉A|eji 〉B

are the eigenvalues and eigenvectors of the operator∑

i pi|i〉〈i| ⊗ ρi,which means that:

S

(∑i

pi|i〉〈i| ⊗ ρi

)= −

∑ij

piλji log piλ

ji = H(pi) +

∑i

piS(ρi).

4. Suppose pi are probabilities, the states ρi have support on orthogonalsubspaces of a space HA, and σi have support on orthogonal subspacesof another space HB. Similarly to the last two proofs, let piλ

ji and

|eji 〉A be the eigenvalues and corresponding eigenvectors of piρi, while

ηki and |gki 〉B are the eigenvalues and eigenvectors of σi. Hence piλjiηki

are the eigenvalues of the operator∑

i ρi ⊗ σi corresponding to the

eigenvectors |eji 〉A|gki 〉B. Therefore by the definition of von Neumann

27

entropy:

S

(∑i

piρi ⊗ σi

)= −

∑ijk

piλjiηki log(piλ

jiηki )

= −∑i

pi log pi −∑i

pi∑jk

λjiηki log(λjiη

ki )

= H(pi)−∑i

pi

∑j

λji log λji +∑k

ηki log ηki

= H(pi) +

∑i

pi (S(ρi) + S(σi)) (3.19)

By calling Γi ≡ ρi ⊗ σi and using property 2 one has that:

S

(∑i

piρi ⊗ σi

)= S

(∑i

piΓi

)= H(pi) +

∑i

piS(Γi) (3.20)

Now comparing (3.19) with (3.20) one has that:

S(Γi) = S(ρi) + S(σi) (3.21)

which proves the theorem.

More important properties can be deduced once one defines ([3]):

• Joint entropy for a composite system with two components A andB in the obvious way: S(A,B) = −Tr(ρAB log ρAB) where ρAB is thedensity matrix of the system AB.

• Conditional entropy by:

S(A|B) = S(A,B)− S(B).

• Mutual information by:

S(A : B) = S(A)+S(B)−S(A,B) = S(A)−S(A|B) = S(B)−S(B|A).

Some properties of the Shannon entropy H(X) fail to hold for the von Neu-mann entropy, for instance for random variables X and Y , the inequalityH(X) ≤ H(X,Y ) holds. This makes intuitive sense: surely we cannot bemore uncertain about the sate of X than we are about the joint state of Xand Y . This intuition fails for quantum states. Consider a system AB oftwo qubits in the entangled state Φ+. This is a pure state, so S(A,B) = 0.On the other hand, system A has density operator I/2, and thus has entropyequal to one. Another way of stating this result is that, for this system, thequantity S(B|A) = S(A,B)− S(A) is negative. In other words, the subsys-tems of the entangled system may exhibit more disorder than the system asa whole. In the classical world this never happens.

28

Property 5 Suppose |AB〉 is a pure state of a composite system. Then|AB〉 is entangled ⇐⇒ S(B|A) < 0.

Property 6 (Subadditivity) Suppose distinct quantum systems A and Bhave a joint state ρAB. Then the joint entropy for the two systems satisfiesthe inequalities:

S(A,B) ≤ S(A) + S(B) S(A,B) ≥ |S(A)− S(B)|.

Property 7 (Concavity) The von Neumann entropy is a concave functionof its inputs. That is, given probabilities pi – non-negative real numbers suchthat

∑i pi = 1– and corresponding density operators ρi, the entropy satisfies

the inequality:

S

(∑i

piρi

)≥∑i

piS(ρi).

(note that in this case we didn’t make any assumption on the operators ρi,contrarily to property 2).

Proof

5. S(A,B) = S(B,A) = 0 since |A,B〉 is pure. If |AB〉 is entangled,then ρA is a mixed state, for which S(A) > 0, consequently S(B|A) =S(B,A)− S(A) = −S(A) < 0.

If S(B|A) < 0 then S(A) > 0 indicating that ρA is a mixed state,which means that |A,B〉 is entangled.

6. This proof has been omitted.

7. Suppose the ρi are states of a system A. Introduce an auxiliary systemB whose state space has an orthonormal basis {|i〉} corresponding tothe index i on the density operators ρi. Define a joint state of AB by:

ρAB =∑i

piρi ⊗ |i〉〈i| (3.22)

To prove concavity we use the sub-additivity of the entropy. Note thatfor the density matrix ρAB we have:

S(A) = S(ρA) = S(TrB(ρAB)) = S

(∑i

piρi

)(3.23)

S(B) = S(ρB) = S(TrA(ρAB)) (3.24)

= S

(∑i

pi|i〉〈i|

)= (property 2) = H(pi)

S(A,B) = H(pi) +∑i

piS(ρi) (thanks to property 3)

29

Applying the sub-additivity inequality S(A,B) ≤ S(A) + S(B), weobtain: ∑

i

piS(ρi) ≤ S

(∑i

piρi

)which proves the theorem.However we still have to justify equations (3.23) and (3.24). For theformer we can say that:

S(A) = S(ρA) = S(TrB(ρAB)) where

TrB(ρAB) = TrB

(∑i

piρi ⊗ |i〉〈i|

)=

(we choose basis {|i〉} to compute the partial trace)

=∑k

∑i

piρi〈k|i〉〈i|k〉 =∑k

∑i

piρiδik2 =

=∑i

piρi.

For the latter:

S(B) = S(ρB) = S(TrA(ρAB)) where

TrA(ρAB) = TrA

(∑i

piρi ⊗ |i〉〈i|

)=

we choose the eigenbasis {|eki′〉} ⊂ HA of a fixed operator ρi′ , with cor-responding eigenvalues λki′ , to compute the partial trace over systemA:

=∑ik

〈eki′ |piρi|eki′〉|i〉〈i| =

=∑k

∑i 6=i′〈eki′ |piρi|eki′〉|i〉〈i|+

∑k

λki′pi′ |i′〉〈i′|

=∑k

∑i 6=i′〈eki′ |piρi|eki′〉|i〉〈i|+ pi′ |i′〉〈i′|

After iterating this procedure N times (N being the number of theoperators ρi and thus the dimension of the auxiliary system B) andeach time choosing an eigenbasis of a different ρi, we then add all leftsides of the equations obtained as well as the right sides:

NTrA(ρAB) =

N∑i

pi|i〉〈i|+

(NTrA(ρAB)−

∑k

∑i

〈eki′ |piρi|eki′〉|i〉〈i|

)

=

N∑i

pi|i〉〈i|+ (N − 1)TrA(ρAB)

30

which yields

TrA(ρAB) =∑i

pi|i〉〈i|

that concludes the proof.

Entanglement distillation and dilution. In order to give a physicalinterpretation to the entropy of entanglement, we will now define two keyconcepts in the context of quantum information: entanglement distillationand dilution ([3] and [6]). As far as the former is concerned, let us considerthe state (3.14) and let us assume that our goal is to create a maximallyentangled state (i.e. a state in the same form, but with θ = π/4) by actinglocally on each of the particles. In order to do that, we can try to performa generalized measurement on the first particle. Hence we choose {A0, A1}as our collection of measurement operators acting on the first particle, with

A0 = tan(θ)|0〉〈0| + |1〉〈1| and A1 = (1−A0†A0)

1/2. Those two operators

fulfill (2.13). In case we measure and obtain the outcome associated withA0 we will achieve our goal; as a matter of fact, the state of the system afterthe measurement is the Bell state Φ+:

ρ0 = (recall chapter 2) =A0ρA0

†

p0= ... = |Φ+〉〈Φ+|.

where ρ = |Φ〉〈Φ| and p0 is our probability of succeeding: p0 = Tr(A0ρA0†) =

... = 2 sin2(θ) (note that θ ∈ (0, π/4)). On the contrary, if we obtain theoutcome associated with A1, we have produced a product state. We pointout that, if A and B are spatially separated and are held by Alice and Bob,respectively, in order to know if the measurement has been successful, theoutcome (i.e. classical bit of information) has to be transmitted from Al-ice to Bob. One says that by local operations and classical communication(LOCC) one can distill a maximally entangled state out of the state Φ withprobability p0. One may wonder if there is another generalized measurementapplied to A and B (individually) giving a higher probability of success. Infact, this is not the case since the generalized measurement we just choseis the optimal one. Now one can consider the case in which Alice and Bobpossess two identical copies of the state Φ, and they try to obtain maximallyentangled states by LOCC (in which joint measurements on both qubits ofAlice, or both qubits of Bob, are authorized). In general, they may obtain asoutcome a maximally entangled state in a Hilbert space – of entangled states– of dimension d = 2, 3, 4. For instance, if they are completely successful,they will get two copies of a maximally entangled state –that is a Bell state–which is equivalent to a single copy of such a state in a space of dimensiond = 4. Or if they get a single copy, they will have d = 2 (for example, theHilbert spaces {|Φ+〉, |Φ−〉} or {|Ψ+〉, |Ψ−〉}). Now we can consider whathappens when we take n copies and allow for the optimal LOCC in order to

31

optimize

E =1

n

2n∑d=1

pd log2(d)

where pd is the probability they end up with an entangled state in a spaceof dimension d (for d = 1 they end up in a product state). The logarithm isthe right quantity such that n copies of a maximally entangled state (whichcorresponds to a dimension dn) exactly give a factor n. In this case Aliceand Bob try to maximize quantity E (note that E ≤ 1)) in order to obtainas much Bell states as possible out of the n copies of state Φ. It turnsout that in the limit n→∞, the result, called entanglement of distillation,precisely coincides with the entropy of entanglement E(Φ). This occursnot only for qubits, but for any d-level systems. Thus, the entanglemententropy is nothing but the optimal (i.e. maximized) averaged amount ofentanglement (i.e. of Bell states) that we can distill out of Φ by LOCCin the asymptotic limit where we have a large number of copies. In otherwords, the physical purpose of entanglement entropy E(Ψ) is to quantifythe amount of entanglement contained in the pure state |Ψ〉.

One may consider the opposite process, called entanglement dilution:given n maximally entangled states, and by applying the optimal LOCC,how many copies, m, of a state |Ψ〉 we can obtain. The ratio n/m in thelimit n→∞ is called entanglement of formation and turns out to coincideagain with E(Ψ), giving another physical meaning to the latter.

Concurrence and Fidelity. We finish this section by mentioning otherquantities that are usually employed to quantify entanglement. One is con-currence, which in the case of pure states reduces to the product of theSchmidt coefficients. Another one is the fidelity with a maximally entangledstate:

F (Ψ) = max |〈Φ+|(UA ⊗ VB)|Ψ〉|2

where the maximization is with respect to the unitary operators U and V .It measures in a sense how close we are to a maximally entangled state; Uand V just correspond to a basis change.

3.2.2 Multipartite systems

Entanglement in multipartite systems becomes more complicated than inbipartite ones. First of all, one can have a situation where certain objectsare entangled to others, but not to all of them. Second, the quantificationgets harder, since it is not known if a property like the inter-convertibilityof states by distillation and dilution exists [3].

Definition 2 We say that a state Ψ of systems A,B, ...Z is a product state

32

if there exist |ϕX〉 ∈ HX such that

|Ψ〉 = |ϕA〉 ⊗ |ϕB〉 ⊗ ...⊗ |ϕZ〉

otherwise we say that we have an entangled state. This definition was givenby Schrodinger.

Although a pure state Ψ is entangled, it may happen that some of thesubsystems are still disentangled. In order to characterize the entanglement,we consider all possible partitions of the subsystems and for each of themwe apply the above definition, with an element of the partition replacing asingle subsystem (note that an element of the partition can contain morethan one subsystem). For instance, for three parties we can have: (i) theyare in a product state; (ii) only A and B are entangled; (iii) only A and Care entangled; (iv) only B and C are entangled; (v) all are entangled. Thesecases are mutually disjoint. An example of case (i) is |0, 0, 0〉, of case (ii)|Φ+〉AB ⊗ |0〉C , and of case (iii) the states:

|W 〉 =|0, 0, 1〉+ |0, 1, 0〉+ |1, 0, 0〉√

3

|GHZ〉 =|0, 0, 0〉+ |1, 1, 1〉√

2

These two last examples illustrate the difficulty of quantifying entanglementin many-body quantum systems; it is not clear which of those states is“more” entangled. Furthermore, in this case it is not possible to convert thestate |W 〉 into the state |GHZ〉 (or vice versa) by LOCC in the asymptoticlimit without losing copies (equivalently, entanglement), and thus we cannotassign a quantity like the entanglement entropy to them.

What we can still do is consider bipartite partitions, in which we computethe entropy of entanglement of two disjoint sets of subsystems. Anotherpossibility is to look at fidelities with respect to certain particular states.For instance, one can define the fidelity with respect to a GHZ or to a Wstate; or to products of Bell states.

Moreover, along with the problem of quantifying entanglement, there isalso the problem of its detection: while in bipartite systems SD allows us toimmediately detect whether a pure state is entangled, in multipartite onesSD is typically not available.

3.3 Mixed States

Pure states are hardly representative of the majority of physical situations.As a matter of fact, a system very soon interacts with a number of systemsso that, even if it was prepared in a pure state, it is typically described bya mixed state characterized by a certain probability of finding the above

33

mentioned state within the pure state in which it was prepared. Hencethe definition of entanglement has broadened beyond Schrodinger’s originaldefinition in order to include such mixed states [2].

3.3.1 Bipartite systems

We consider ([6]) again two subsystems, A and B.

Definition 3 We say that ρ represents a product state whenever we canfind ρA, ρB operators acting on HA,HB such that:

ρ = ρA ⊗ ρB

Definition 4 A state ρ is separable if it can be written as a statisticalmixture of product states; that is, if it can be written as a convex combinationof products of subsystem states:

ρ =∑i

piρAi ⊗ ρBi

Without loss of generality, we can assume that ρAi and ρBi are all pureensembles of the appropriate subsystems, i.e. ρAi = |a〉〈a|i and ρBi = |b〉〈b|i;indeed, if they were mixed states, we could make them explicit through (1.16)obtaining a sum, over 3 indexes, of pure states and we could then reduce suchsum to only one index:

ρ =∑i

pi|a〉〈a|i ⊗ |b〉〈b|i =∑i

pi|aibi〉〈aibi|

Otherwise, we say that ρ represents an entangled state.

Note that while every mixed state can be written as:

ρ =∑i

pi|Ψi〉〈Ψi|

requiring that ρ is separable means that every operator |Ψi〉〈Ψi|, which actson HAB, is a product state: |Ψi〉〈Ψi| = ρAi⊗ρBi, or equivalently that everystate |Ψi〉 is a product state |ai〉|bi〉. However it is not always possible totell whether or not a given mixed state is separable; this problem is knownas the separability problem.

What differentiates pure and mixed states is that non-entangled mixedstates may contain correlations, while non-entangled pure states do not. Forexample, the mixed separable state:

ρ =1

2(|0〉〈0| ⊗ |0〉〈0|+ |1〉〈1| ⊗ |1〉〈1|) =

1

2(|0, 0〉〈0, 0|+ |1, 1〉〈1, 1|)

34

fulfills 〈σAz ⊗σBz 〉 = Tr(σAz ⊗σBz ρ) = ... = 1 whereas 〈σA,Bz 〉 = Tr(σA,Bz ρA,B) =... = 0. These correlations are, however, very trivial. If we had a classicalsystem we could also have them. Only entangled states may display non-classical correlations.

A subtle point is that a state may look entangled even though it isseparable. Let us take, for instance:

σ =1

2(|Φ+〉〈Φ+|+ |Φ−〉〈Φ−|)

According to this formula, we can prepare σ by mixing two maximally en-tangled states. However, there is another way of preparing the same stateσ which does not require using entangled states at all. This immediatelyfollows from the fact that σ = ρ (one has just to replace the definition ofthe Bell states in this formula), and thus one can prepare it by mixing twoproduct states. Therefore σ is a separable state as well. This simple exam-ple illustrates the difficulty of finding out whether a state is entangled ornot: we have to check all the possible decompositions and only if none ofthem involves exclusively product states, we will have an entangled state.Unfortunately there exist infinite decompositions, so that this task is hope-less. Fortunately, in some special cases there are shortcuts, which can giveus the right answer with much less effort. We will now go through them.

Entanglement witnesses. An entanglement witness is an observablewhich detects the presence of entanglement. Given an operator W = W † wesay that it is a witness if for all product states |a, b〉 we have 〈a, b|W |a, b〉 ≥ 0,butW possesses negative eigenvalues. From the definition of separable statesit is clear that if Tr(ρW ) < 0, then ρ must be entangled; as a matter of factif ρ is separable, then Tr(ρW ) ≥ 0:

Tr(ρW ) =∑kl

〈kl|∑i

pi|aibi〉〈aibi|W |kl〉 =

(being {|kl〉} an orthonormal basis in HAB)

= (writing |aibi〉 in such basis: |aibi〉 =∑

nm c(i)nm|nm〉)

=∑kli

pi∑nm

〈kl|c(i)nm|nm〉c(i)

nm

∗〈nm|W |kl〉 =

=∑klinm

|c(i)nm|

2piδknδlm〈nm|W |kl〉 =

=∑kli

pi|c(i)kl |

2〈kl|W |kl〉 ≥ 0

since pi ≥ 0, |c(i)kl |

2≥ 0, 〈kl|W |kl〉 ≥ 0. Thus, a negative expectation value

of a witness indicates the presence of entanglement. Note, however, thatthe converse is not necessarily true: if the expectation value of a witness is

35

positive, this does not imply that the corresponding state is not entangled.Hence the important question one can ask about entanglement witnessesregards their optimality. We say that an entanglement witness W1 is finerthan W2 if and only if the entanglement of any ρ detected by W2 is alsodetected by W1. Horodecki showed ([5]) that for any entangled state therealways exists a witness that detects it. Sadly, there is no simple way of find-ing out such a witness, which makes the problem of detecting entanglementin mixed states rather non-trivial.

Let’s consider the following example. For a system of two qubits, W =2 − S, where S = σA1 ⊗ (σB1 + σB2 ) + σA2 ⊗ (σB1 − σB2 ) and the sigma’s arePauli operators, is an entanglement witness. This can be shown by noticingthat for a product state: |〈S〉| ≤ |〈σB1 〉 + 〈σB2 〉| + |〈σB1 〉 − 〈σB2 〉| ≤ 2, since〈σ〉 ≤ 1. By choosing the sigma’s as follows: σ1 = σ~n, σ2 = σ~m (where σ~n,~mare those in subsection (3.2.1)), we see that for ρ(p) = (1−p)1

4 +p|Φ+〉〈Φ+|,entanglement is detected for p > 1√

2.

PPT criterion. A very strong necessary condition for separability hasbeen proved by Peres, called the positive partial transpose (PPT) criterion(more details in [5]). It says that if ρAB is separable, then the new operatorρTBAB, with matrix elements defined in some fixed product basis as

(ρTBAB)mµ,nν = 〈m|〈µ|ρTBAB|n〉|ν〉 ≡ 〈m|〈ν|ρAB|n〉|µ〉

is a density operator (i.e. has a non-negative spectrum), which means thatρTBAB is also a quantum state. Thus, if ρTBAB has any negative eigenvalue,then ρAB must necessarily be entangled. It also guarantees the positivity ofρTAAB defined in an analogous way. The operation TB, called a partial trans-pose, corresponds to transposition of indexes corresponding to the secondsubsystem. A fundamental fact pointed out by Horodecki is that the PPTcondition is not only a necessary but also a sufficient condition for separa-bility of the 2⊗ 2 and 2⊗ 3 cases. Thus it gives a complete characterizationof separability in those cases.We will briefly prove the PPT criterion:

if ρAB is separable, then it can be written

ρAB =∑i

piρAi ⊗ ρBi

In this case, the effect of the partial transposition is trivial:

ρTBAB =∑i

piρAi ⊗ ρBTi

As the transposition map preserves eigenvalues –the eigenvaluesof an operatorA are equal to the eigenvalues of its transpose since

36

they share the same Characteristic polynomial: det(A − λI) =det (A− λI)T = det(AT −λI)– the spectrum of ρTBAB is the same

as the spectrum of ρAB, and in particular ρTBAB must still bepositive semidefinite, which concludes the demonstration.

Entanglement measures and mutual information. For mixed states,entanglement measures similar to entanglement entropy can be defined, al-though they are much harder to evaluate. For instance, we can considerthe distillation procedure as before, but now with mixed states. That is,we may try to distill out of n copies of a state ρ the maximal number m ofBell states using LOCC. The ratio m/n in the limit n→∞ is called entan-glement of distillation D(ρ). Analogously, we may consider the process ofentanglement dilution and define the entanglement cost Ec(ρ). In general,D(ρ) < Ec(ρ) so that we cannot first distill and then get back the same stateas before [6]. Moreover, there exist very few examples where these quantitiescan be evaluated. One could wonder why entanglement cannot be evaluatedby the entropy of entanglement –that is the von Neumann entropy of thereduced density operator – in the case of bipartite mixed states. In orderto answer this question, let’s list the axioms that a satisfactory measure ofentanglement over mixed systems E(ρ) should fulfill ([4]):

Axiom 1 E(ρ) reduces to the entropy of entanglement ES(ρ) in case ρ is apure state.

Axiom 2 E(ρ) = 0 for separable states.

Axiom 3 A measure of entanglement is not increasing under LOCC:E(UρU−1) ≤ E(ρ). Since a measure of entanglement gives a measure aboutthe amount of quantum correlations between two subsystems, it cannot in-crease if we modify a subsystem (LO) or if the two subsystems exchangeclassical information (CC). Focusing on local operations, since they are im-plemented by unitary operators, we can express this by writing: E(UρU †) ≤E(ρ).

Axiom 4 (Convexity) Entanglement should not increase by mixing var-ious density matrices, since a convex combination of density matrices isa purely classical statistical superposition of states: E(λρ1 + (1 − λ)ρ2) ≤λE(ρ1) + (1− λ)E(ρ2).

Axiom 5 Similarly entanglement should not increase if we simply put to-gether two states describing different subsystems: E(ρ⊗ σ) ≤ E(ρ) +E(σ).

Comparing these properties with the ones fulfilled by the von Neumannentropy, one immediately notices that while S(ρ) is concave, E(ρ) is con-vex. This striking difference is basically the reason for which the entropy

37

of entanglement fails to hold as a satisfactory measure of entanglement overmixed states.This statement can be better visualized by considering a generic mixed stateρ: ρ =

∑i pi|Ψi〉〈Ψi| and by evaluating its entanglement through the en-

tropy of entanglement:

ES(ρ) = S(ρA) where

ρA = TrB(ρ) =∑i

piTrB(|Ψi〉〈Ψi|) ≡∑i

piρAi

⇒ S(ρA) = S

(∑i

piρAi

)and due to the concavity of the von Neumann entropy:

ES(ρ) = S(ρA) >∑i

piS(ρAi) (3.25)

(equality occurs only when all ρAi are equal). The last inequality is clearlyin contradiction with the axiom of convexity:

E(ρ) = E

(∑i

pi|Ψi〉〈Ψi|

)≤∑i

piE(Ψi)

recalling axiom 1: E(Ψi) = ES(Ψi) = S(ρAi), which yields

E(ρ) ≤∑i

piS(ρAi) (3.26)

Inequalities (3.25) and (3.26) show that ES(ρ) 6= E(ρ), hence the entropyof entanglement can not be taken as a measure of entanglement for mixedstates.

A possible measure of entanglement that satisfies the above axioms and thatcan be determined in practice is the entanglement of formation:

EF (ρ) = min∑i

piE(Ψi)

where E(Ψ) is the entropy of entanglement and the minimization is donewith respect to all decompositions of ρ. This quantity is related to theentanglement cost through EF (ρ⊗n)/n → Ec(ρ) in the limit n → ∞. Notethat in the pure state case, E(Ψ) = EF (Ψ) = Ec(Ψ) = D(Ψ).

Another way of measuring entanglement is through the fidelity with amaximally entangled state, as the one already defined, but now with:

F (ρ) = max〈Φ+|(U ⊗ V )ρ(U † ⊗ V †)|Φ+〉 =∑i

piF (Ψi)

38

where F (Ψ) is the fidelity previously defined and pi are the coefficients ofthe convex combination representing ρ.

Finally, entanglement can also be measured using the definition of partialtransposition. One defines negativity as

N(ρ) = max(‖ρTA‖1 − 1, 0

)(3.27)

where the 1-norm is given by the sum of the absolute values of the eigen-values. Negativity can be positive only if we have an entangled state, so it’sa sufficient condition but it is not necessary: there exist entangled statesfor which it is zero. In fact, we are going to prove that N(ρ) > 0 ⇒ ρ isentangled:

N(ρ) > 0⇒ ‖ρTA‖1 − 1 > 0⇒ ‖ρTA‖1 > 1. (3.28)

Now since Tr(ρ) = 1 and

Tr(ρTA) =∑mµ

〈m|〈µ|ρTA |m〉|µ〉 =

=∑mµ

〈m|〈µ|ρ|m〉|µ〉 =

= Tr(ρ)

We have that Tr(ρTA) = 1⇒∑

k λk = 1 (being ρTA |k〉 = λk|k〉).Now considering (3.28) we have that

∑k |λk| > 1, which com-

pared with the previous equation means that there exists at leastone k for which λk < 0, thus ρTA has at least a negative eigen-value. Applying the PPT criterion we conclude that ρ is anentangled state.

Another quantity of interest is the already known quantum mutual informa-tion S(A : B) = S(A) + S(B) − S(A,B). This does not measure entangle-ment, but rather correlations. In fact, it is the finest measure of correlations,in the sense that it detects them even when correlation functions do not [6].It satisfies the following properties:

1. S(A : B) ≥ 0; S(A : B) = 0⇔ ρ = ρA ⊗ ρB

2. S(A : B) ≤ S((aA) : B) ≤ S(A : B) + 2S(a)

The first indicates that it is zero only for product states, i.e. when there areno correlations. The second indicates that it decreases whenever we discarda subsystem (in this case a), but it cannot decrease by more than twice theentropy of such a system.Proof

39

1. At first let’s show that there are no correlations for product statessuch as ρ = ρA ⊗ ρB:

〈A⊗B〉 = Tr(ρA⊗B) =∑kl

〈kl|ρA⊗B|k〉|l〉 =

=∑kl

〈kl|ρA ⊗ ρB|A|k〉B|l〉 =

=∑kl

〈kl|ρAA|k〉ρBB|l〉 =

=∑kl

〈k|ρAA|k〉〈l|ρBB|l〉 =

= Tr(ρAA)Tr(ρBB) =

obviously for product states the reduced density matrix 3 for systemA,B is ρA,B, resulting in:

〈A⊗B〉 = Tr(ρAA)Tr(ρBB) = 〈A〉〈B〉 (3.29)

which confirms the absence of correlations.In order to prove the property we will use the so called strong sub-additivity of the von Neumann entropy: given three subsystemsX,Y, Zthen

S(X,Y ) + S(X,Z) ≥ S(X,Y, Z) + S(X)

If we choose Y = A,Z = B and X a system which is pure (S(X) = 0)and disentangled from the others (i.e. XY and XZ are product states)we immediately obtain the first property, since S(X,Y ) = S(ρX ⊗ρY ) = recall the von Neumann property number 4 = S(X) + S(Y ) =S(Y ) and S(X,Z) = S(ρX⊗ρZ) = S(X)+S(Z) = S(Z). The equalityholds when ρ = ρA ⊗ ρB: S(A,B) = S(ρA ⊗ ρB) = S(ρA) + S(ρB) =S(A) + S(B).

2. By choosing X = A, Y = B and Z such that the whole state Ψ ispure and the reduced state in AB is our state ρ, we obtain S(A,B) ≥S(A)−S(A,Z); using property 1 of the von Neumann entropy we haveS(A,Z) = S(B) (being ABZ a pure state), which yields: S(A,B) ≥S(A)− S(B).We prove S(A : B) ≤ S((AZ) : B), which is equivalent to S(A) +S(B) − S(A,B) ≤ S(A,Z) + S(B) − S(Z,A,B), by simply choosingX = A, Y = B in the strong sub-additivity inequality.We prove S((AZ) : B) ≤ S(A,B)+2S(Z) making it explicit: S(A,Z)+S(B) − S(A,Z,B) ≤ S(A) + S(B) − S(A,B) + 2S(Z) and using in-equalities: S(A,Z,B) ≥ S(A,B) − S(Z) (obtained from S(A,B) ≥S(A)− S(B)) and S(Z) + S(A) ≥ S(A,Z) (obtained from property 6of the von Neumann entropy).

3TrB(ρ) = TrB(ρA ⊗ ρB) = ρATrB(ρB) = ρA

40

Moreover, quantum mutual information for pure states Ψ reduces to theentropy of entanglement: S(A,B) = 0 and S(A) = S(B), which yield:S(A : B) = 2S(A) = 2E(Ψ).

3.3.2 Multipartite systems

The description of multipartite entanglement must still confront several chal-lenges for mixed states. On the one hand, the definitions of product andseparable states are straightforward:

Definition 5 The state ρA1...Am of m subsystems A1, . . . , Am is a productstate whenever we can find ρA1 , . . . , ρAm operators acting on HA1 , . . . ,HAm

such that:ρA1...Am = ρA1 ⊗ ...⊗ ρAm

Definition 6 The state ρA1...Am of m subsystems A1, . . . , Am is completelyseparable if and only if it can be written in the form:

ρA1...Am =∑i

piρA1 i ⊗ . . .⊗ ρAm i

On the other hand we have to consider again different partitions in order tocharacterize entanglement, since it may happen that some of the subsystemsare disentangled, even if the whole state is entangled.

Definition 7 The state ρA1...Am of m subsystems A1, . . . , Am is entangledwith respect to a given partition {I1, . . . , Ik}, where Ii are disjoint subsetsof the indices I = {1, . . . ,m}, ∪kj=1Ij = I, if and only if it can not be writtenin the form:

ρA1...Am =∑i

piρ1i ⊗ . . .⊗ ρki

We end up with a table in which for each partition we declare whether thestate is entangled or not. In order to check the entanglement for each parti-tion, we have to, for example, find the appropriate witness, whose definitionfollows very naturally the one for bipartite systems. Such a table may con-tain some redundancies since, for instance, if a tripartite state is completelyseparable, i.e. separable with respect to the partition (A)(B)(C), it is auto-matically separable for any other partition, and thus not entangled at all.This is obviously a general fact: if a multipartite state is completelyseparable, then it is not entangled with respect to any possiblepartition.

41

3.4 An interesting example

In this section we will show that the von Neumann entropy and the statefunction entropy known in thermodynamics have something in common. Inparticular, we will compute ([8, page 188]) the von Neumann entropy of anensemble in thermal equilibrium and show ([4]) that it coincides with theabove-mentioned thermodynamic entropy.

First we obtain the density operator ρ describing the ensemble in thermalequilibrium. The basic assumption we make is that nature tends to maximizethe von Neumann entropy of the ensemble S(ρ), subject to the constraintthat the ensemble average of the Hamiltonian has a certain prescribed value,since it is in equilibrium. Once thermal equilibrium is established, we expect:

dρ

dt= 0 (3.30)

otherwise 〈H〉 = Tr(ρH) would be time-dependent, in contradiction withthe hypothesis of equilibrium. Because of (1.27), this means that ρ andH can be simultaneously diagonalized. Therefore, the eigenstates |k〉 usedto trace: S(ρ) = −

∑k xk log xk, which are ρ’s eigenstates (and xk are its

eigenvalues), are chosen to be also energy eigenstates. With this choice, thefractional population of the state |k〉 expresses the probability of finding amixture element with energy Ek.

Thus we are trying to maximize S(ρ), taking into account the constraints:

• The ensemble average of H has, at equilibrium, a prescribed value U :〈H〉 = Tr(ρH) = U , which yields:

Tr(ρH)− U =∑k

xkEk − U = 0 (3.31)

• For any density operator its eigenvalues sum to one:∑k

xk − 1 = 0 (3.32)

We can most readily accomplish this by using Lagrange multipliers, whereS(ρ) is a function of N variables xk and the two constraints (3.31) , (3.32)introduce two Lagrange multipliers, which we call λ1 = β and λ2 = γ. Weobtain:

log xk + 1 + βEk + γ = 0 ⇒ xk = e−βEk−γ−1

The constant γ can be eliminated using (3.32):

∑k

e−βEk−γ−1 = 1 ⇒ Z ≡N∑k

e−βEk = eγ+1

42

where we recognize the partition function Z, that can also be written Z =Tr(e−βH); in conclusion

xk =e−βEk

Z(3.33)

where the constant β can also be determined in terms of U through the firstconstraint.

Had we attempted to maximize S(ρ) without the internal-energy con-straint, we would have obtained instead:

xk =1

N(3.34)

which is the density matrix eigenvalue appropriate for a completely randomensemble, as we stated in section 1.7. Comparing (3.33) with (3.34), weinfer that a completely random ensemble can be regarded as the β → 0limit (physically the high-temperature limit) of a canonical ensemble.

Now, since ρ and H are simultaneously diagonalized with respect to basis{|k〉}, one can write ρ through its spectral decomposition:

ρ =∑k

xk|k〉〈k| =∑k

e−βEk

Z|k〉〈k| =

=∑k

e−βH

Z|k〉〈k| = e−βH

Z

∑k

|k〉〈k| =

=e−βH

Z(3.35)

The last expression for ρ, with β = 1/kBT , is known in statistical mechanicsto describe a canonical ensemble.

Now it’s time to compute the von Neumann entropy of the canonical ensem-ble:

S(ρcan) = −Tr

(e−H/kBT

Zlog

e−H/kBT

Z

)where log e−H/kBT

Z = log e−H/kBT − logZ and log e−H/kBT = −H/kBT as faras the exponent is a Hermitian operator. Thus:

S(ρcan) =1

kBTTr(ρcanH) + logZTr(ρcan) =

recalling that Helmholtz free energy F = −kBT logZ

=〈H〉 − FkBT

recalling F ’s definition: F = U − TStherm and that 〈H〉 = U

= Stherm/kB

We have shown that, up to a factor, the von Neumann entropy of a canonicalensemble is the state function entropy in the thermodynamic sense!

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3.5 Monogamy and Area Law

In this section we will present, from a qualitative point of view, two interest-ing properties fulfilled by many-body quantum systems: the area law ([6])and the monogamy of entanglement ([4]).

The area law seems to be satisfied by all systems in thermal equilibriumwhich interact with short range interactions in lattices.For instance, we consider a spin system on a lattice in d spatial dimensions.The system at temperature T is described by the density operator:

ρ =e−H/kBT

Z

where the Hamiltonian H expresses the short range interaction involvingonly a few spins close to each other. The area law can be obtained both atzero and finite temperature.

In the first case, the state ρ reduces to a projector onto the groundsubspace, whose vector basis {|Ψi〉} fulfills H|Ψi〉 = E0|Ψi〉 where E0 is theground state energy. If the ground state is not degenerate, which we willassume, we will just use a pure state |Ψ0〉 to denote the ground state. Weconsider a connected region A of the lattice with a smooth boundary and thecomplementary region B, and concentrate on the amount of entanglementbetween these two regions, and its growth as we make region A larger andlarger. Since we are dealing with a bipartite system of a pure state |Ψ0〉,we can evaluate such amount through the entropy of entanglement E(Ψ0) =S(ρA). In general, since entropy is an extensive quantity, one would expectthat it scales with the number of spins in region A. However, for groundstates of Hamiltonians as we are considering here, this seems not to be thecase. Instead, the entanglement (entropy of entanglement) scales not withthe volume of region A, but with its boundary area, and thus the name arealaw.

In the second case, the one with finite temperature, ρ is in general abipartite mixed state, hence its amount of entanglement cannot be mea-sured by the entropy of entanglement (see discussion in subsection 3.3.1).Nevertheless, one can still derive an analogous area law by measuring theamount of correlations between regions A and B through the quantum mu-tual information S(A : B). Note that at zero temperature, i.e. pure states,the quantum mutual information reduces (up to a factor two) to the entan-glement entropy, providing us with a suitable generalization of the previousarea law.

Monogamy is one of the most fundamental properties of entanglement andfurther differentiates the notion of entanglement from classical correlations.In many-body systems with local interactions, classical correlations typically

44

decay with distance, but there’s no upper bound to the total number ofcorrelations a single particle can establish with the others of the system. In-stead, in 2006 Osborne and Verstraete proved the Coffman-Kundu-Wootters(CKW) monogamy inequality, which states that given a multipartite systemwith subsystems (qubits) A,B1, B2, . . . then it results:

C2(A|B1) + C2(A|B2) + . . . ≤ 4∆A ≤ 1 (3.36)

where C2(A|Bk) is the concurrence of the bipartite mixed system of qubitsA|Bk (concurrence can also be defined for mixed states, and it still representsa measure of entanglement as in the pure case); ∆A is the determinant ofthe reduced density matrix ρA and it quantifies the entanglement betweenA and the rest of the system.

It appears clearly that it is not possible for A to be strongly entangledwith several subsystems B. If, for instance, A is strongly entangled with B1

then necessarily it can not be equally entangled, nor entangled at all, withthe other subsystems. This is the reason why it is said that the entanglementbond is monogamous and why it differs from what happens in classicalcorrelations.

45

Chapter 4

Conclusions

In this paper we have briefly given insight into the world of entanglementand entanglement entropy.In the first chapter we introduced the notion of density operator and ana-lyzed its properties in order to show, in the following chapter, how QuantumMechanics could be reformulated in the density operator picture. As wehighlighted, this new formulation is equivalent to the most-known one; how-ever, the density operator approach really shines for two applications: thedescription of quantum systems whose state is not known, and the descrip-tion of subsystems of a composite quantum system. Since the entanglementof a system arises as a quantum correlation between two or more subsystems,the density operator picture is fundamental in its description.

In the third chapter we presented the basic concepts regarding entangle-ment of bipartite and multipartite systems, both for pure and mixed states.As we have argued, whereas entanglement for bipartite systems is well es-tablished, for multipartite ones there exist many possibilities of definingentanglement measures. It may turn out that one state is more entangledaccording to one measure, but not according to a different one.Nevertheless, we have highlighted two measures of entanglement/correlationswhich have a clear physical meaning. The first one, the entropy of entan-glement, applies to bipartite pure states and is given by the von Neumannentropy of the reduced density operator of one of the subsystems. Thesecond one, the quantum mutual information, measures the correlations be-tween the two subsystems and for pure states it reduces to the entropy ofentanglement. One can apply these two measures to multi-spin systems inlattices by separating all the spins into two disjoint regions which are thenconsidered as a bipartite system. In the particular case where we deal withground or thermal states of short range interaction Hamiltonians, the appli-cation of those measurements gives rise to area laws. These laws state thatthe quantum mutual information, between a region A and its complemen-tary, scales with the number of spins at the boundary of A (and not, as one

46

would expect, with the total number of spins in A).Another peculiar characteristic of entanglement that distinguishes it fromclassical correlations is its monogamy, according to which each subsystemcan be strongly entangled only with one other subsystem, as there is anupper bound to the sum of its quantum correlations.

Entanglement and its quantification have been primary issues in thestudy of black holes by means of quantum field theory and in the scienceof quantum information, for which entanglement is an important resource.Since these two subjects naturally overlap in the context of many-bodysystems (in particular close to criticality), it soon became apparent thatthere is a rich structure to be uncovered by studying entanglement entropyin this context, and by putting together ideas from field theory and quantuminformation science; much has been done, much more is yet to come.

47

Appendices

48

Appendix A

The generic spin-12 state

The most general spin state of a 12 -spin particle is represented, in the eigen-

basis of operator Sz, by the spinor:

χ =

(ab

)= aχ+ + bχ− (A.1)

where a and b are two arbitrary complex numbers. Requiring the normal-ization to one, a and b have to satisfy the relation: |a|2 + |b|2 = 1. Theremarkable fact is that spinor (A.1) represents a state in which the spin isparallel to a precise direction, defined by the versor n = n(ϑ, ϕ), where ϑand ϕ are obtained as follows:

ϑ = 2 arccos |a| = 2 arccos |b| ϕ = arg b− arg a (A.2)

In order to verify the validity of this assertion, let’s start by observing thatit’s not very meaningful to talk about a precise direction defined by thetwo coefficients a and b, since they are probability amplitudes. What wecan actually show is that the spinor χ is an eigenvector of the operatorσn = ~σ · n with corresponding eigenvalue +1. This means that measuring σn(i.e. the spin) along direction n yields the result +1 with 100% probabilityof success; however, being the modulus of the spin vector

√31, we can only

conclude that the spinor is situated on a cone with axis n, and not preciselyalong n. Consequently it is not χ’s direction that is fixed, but rather thedirection on which the projection of the spin vector has the fixed value +1.We could interpret n as the average direction of spinor χ.

Let’s determine the eigenvector of σn and show that it coincides with (A.1).Given the following components for versor n:

nx = sinϑ cosϕ, ny = sinϑ sinϕ, nz = cosϑ

1note that σ2 = 3I

49

and using the explicit form of Pauli’s matrices, it results:

σn =

(cosϑ sinϑe−iϕ

sinϑeiϕ − cosϑ

)Note that σn

2 = I, hence according to Dirac’s theorem the operator σn haseigenvalues λ = ±1. This means that measuring the spin of a 1

2 -particlealong an arbitrary direction n, the only possible results are +h/2 and −h/2.

We define the eigenspinor associated to the eigenvalue λ = 1 as:

χ+n =

(ab

)(A.3)

and impose that:(cosϑ sinϑe−iϕ

sinϑeiϕ − cosϑ

)(ab

)=

(ab

)which implies the following equalities:

a cosϑ+ b sinϑe−iϕ = a

a sinϑeiϕ − b cosϑ = b

and therefore:

b = asinϑ

1 + cosϑeiϕ = a

sin(ϑ/2)

cos(ϑ/2)eiϕ (A.4)

This, substituted in the normalization condition yields:

|a|2(1 + tan2(ϑ/2)) = 1

and thus:|a| = cos(ϑ/2) (A.5)

We then can choose a = cos(ϑ/2) which implies b = sin(ϑ/2)eiϕ thanksto (A.4). In this way, up to an arbitrary phase factor, the spinor (A.3) iscompletely determined:

χ+n = cos(ϑ/2)χ+ + sin(ϑ/2)eiϕχ−

and satisfies (A.2), which ends the proof.

The fact that any spinor like (A.1) represents a spin state with a definitepolarization has an obvious explanation: the spinor is identified by twocomplex numbers, or equivalently four real numbers, but only two of theseare essential in defining the state described by χ. One is eliminated bythe normalization condition, and another is incorporated in an arbitraryphase factor and doesn’t play any role. The two remaining parameters canalways be put in biunivocal correspondence with two polar angles ϑ andϕ, associating to the state a definite polarization direction identified by theversor n = n(ϑ, ϕ) ([1] for further reading).

50

Appendix B

Density Matrix in thePosition operator eigenbasis

A significant representation ([1]) of the density operator ρ is the one givenin the eigenbasis of the position operator x: {δ(x − x′)} with correspond-ing eigenvalues {x′} (supposing that scalar products are computed in theCoordinate space). For simplicity, we will indicate |x〉 the eigenvector ofthe position operator corresponding to eigenvalue x. It is clear that givena pure state |s(i)〉, the scalar product: 〈x|s(i)〉 = ψi(x) outputs the spatialcomponent of that pure state evaluated at x. Given the density operator:

ρ =∑i

pi|s(i)〉〈s(i)|

its representation with respect to the eigenbasis {|x〉} is:

ρxx′ ≡ ρ(x, x′) =∑i

pi〈x|s(i)〉〈s(i)|x′〉 =

=∑i

piψi(x)ψ∗i (x′)

in particular, for diagonal elements we have:

ρ(x, x) =∑i

pi|ψi(x)|2 (B.1)

Diagonal elements coincide with the weighted average of the probabilitydensities, this confirms once again the physical meaning of the fractionalpopulations: they express the probability density of finding a generic elementof the mixture in the state |x〉; in this particular case, the probability densityof finding such generic element in the position x.

51

Appendix C

Shannon and von NeumannEntropy

An entangled pure state is related to mixed subsystems, whose density ma-trices can be written as classical superpositions (1.16) of density matrices ofpure states, with real non-negative statistical weights (the ones defined atthe beginning of section 1.4):

ρ (density operator of a mixed subsystem) =∑k

pkρk (C.1)

whereρk = |k〉〈k|

∑k

pk = 1 0 ≤ pk ≤ 1 ∀k (C.2)

These classical weights define a classical probability distribution, which as-sociates to each state ρk its classical probability pk. This classical nature isevident in the expectation value of an operator O which acts on the subsys-tem considered:

〈O〉 = Tr(ρO) =∑k

pkTr(ρkO) = (ρk = |k〉〈k|)

=∑k

pkTr(|k〉〈k|O) = (Tr(|k〉〈k|O) = 〈k|O|k〉, see (1.9))

=∑k

pk〈O〉k

where the pks play the role of classical weights, since they quantify the weightof each state |k〉 in the mean value of O.

To each classical probability distribution X = {p1, p2, ...pn} we can al-ways associate a function called Shannon Entropy, which measures howmuch a distribution is not deterministic, i.e. the amount of uncertaintyabout X before we learn its value. Shannon Entropy is defined as:

H(X) = H(p1, p2, ...pn) = −∑k

pk ln pk (C.3)

52

(in quantum information theory the natural logarithm is substituted withlog2). For instance, a constant distribution maximizes H(X) since eachoutcome is equally probable, while a delta distribution pk = δk,k0 minimizesH(X) since there’s only one possible issue for a measure. An event that cannever occur should not contribute to the entropy, so by convention we agreethat 0 ln 0 = 0. We could immediately use Shannon Entropy to quantify theuncertainty about the distribution of pure states composing the mixed stateρ, that is, the “amount of mixedness” of such state, hence making Shan-non Entropy a measure of the entanglement of the whole system of which ρrepresents a subsystem. However, since the same operator ρ can be writtenin many different ways –due to the non-uniqueness of the decompositionof mixed states as convex combinations of pure states– we would face theproblem of deciding which decomposition provides the classical probabilitydistribution employed in computing the Shannon Entropy of the selectedmixed state. This problem is solved by defining a generalization of Shan-non Entropy that is suitable for quantum systems, with density operatorsreplacing probability distributions, the von Neumann Entropy :

S(ρ) = −Tr(ρ ln ρ)

(in quantum information theory the natural logarithm is substituted withlog2). Von Neumann Entropy is the quantum counterpart of Shannon En-tropy, in the sense that it measures the mixedness of the state ρ on whichit is applied; as a matter of fact, it coincides with Shannon Entropy whenthe latter is computed on the probability distribution related to ρ’s spectraldecomposition:

ρ =∑k

dk2|k〉〈k| (spectral decomposition)

where the eigenvalues are the square of the Schmidt coefficients;

X = {d12, d2

2, ...dn2}

H(X) = −∑k

dk2 ln dk

2

S(ρ) = −Tr(ρ ln ρ) = (choosing for ρ the diagonalized matrix: δkjdk2)

= −∑k

dk2 ln dk

2

⇒ S(ρ) = H(X)

For instance, the completely mixed density operator in a d-dimensional spacedefined in section 1.7, I/d, has entropy log d.For a more complete study of the properties of the von Neumann entropy,we refer the reader to [3, chapter 11].

53

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