Share this document with a friend

12

Transcript

PHYSICAL REVIEW A 86, 052328 (2012)

Quantum discord for Gaussian states with non-Gaussian measurements

Paolo Giorda,1 Michele Allegra,1,2,3 and Matteo G. A. Paris4,5

1Institute for Scientific Interchange Foundation (ISI), I-10126 Torino, Italy2Dipartimento di Fisica, Universita di Torino, I-10125 Torino, Italy

3INFN, Sezione di Torino, I-10125 Torino, Italy4Dipartimento di Fisica, Universita degli Studi di Milano, I-20133 Milano, Italy

5CNISM, Unita di Milano, I-20133 Milano, Italy(Received 12 July 2012; published 27 November 2012)

In recent years the paradigm based on entanglement as the unique measure of quantum correlations has beenchallenged by the rise of new correlation concepts, such as quantum discord, able to reveal quantum correlationsthat are present in separable states. It is in general difficult to compute quantum discord, because it involves aminimization over all possible local measurements in a bipartition. In the realm of continuous-variable (CV)systems, a Gaussian version of quantum discord has been put forward upon restricting to Gaussian measurements.It is natural to ask whether non-Gaussian measurements can lead to a stronger minimization than Gaussian ones.Here we focus on two relevant classes of two-mode Gaussian states: squeezed thermal states and mixed thermalstates, and allow for a range of experimentally feasible non-Gaussian measurements, comparing the results withthe case of Gaussian measurements. We provide evidence that Gaussian measurements are optimal for Gaussianstates.

DOI: 10.1103/PhysRevA.86.052328 PACS number(s): 03.67.−a, 03.65.Ta

I. INTRODUCTION

In recent years the paradigm based on entanglement [1] asthe unique genuine measure of quantum correlations has beenchallenged by the argument that the notion of nonseparabilitymay be insufficient to encompass all correlations that can befairly regarded as quantum, or nonclassical. This has spurredthe development of conceptually new correlation measures,such as quantum discord [2–4], based on local measurementsand able to reveal quantum correlations that are present evenin separable states. These correlations can be interpreted as anextra amount of information that only coherent operations canunlock [5]. In fact, there are several indications suggesting thatgeneral quantum correlations might be exploited in quantumprotocols [6], including mixed state quantum computation [7]and remote state preparation [8]. Therefore, a more completetheoretical and experimental investigation thereof is now acentral issue in quantum science and technology [5,9–11].

The definition of discord involves an optimization overall possible local measurements in a bipartition, the optimalmeasurement leading to a minimal value of quantum discord.To perform the optimization is remarkably difficult, whichhampers analytical progress in the area. This fact has ledto the definition of other correlation measures which areconceptually similar but easier to compute, such as thegeometric discord [12]. In the realm of finite-dimensionalsystems, where the concept of discord was first introduced,analytic results for quantum (geometric) discord have beenobtained for pairs of qubits when the global state is in X form(in arbitrary form) [12,13].

In the realm of continuous-variable (CV) systems, initialresearch efforts on quantum discord have focused on Gaussianmeasurements. The Gaussian quantum discord, proposed in[14,15], is defined by restricting the minimization involvedin the definition of discord to the set of Gaussian positiveoperator-valued measures (POVMs) [16] and it can be ana-lytically computed for Gaussian states. Its behavior in noisy

channels has been studied in Ref. [17], where it was shownthat it is more robust than entanglement to the decorrelatingeffect of independent baths and more likely to yield non-zeroasymptotic values in the case of a common bath; while its re-lation to the synchronization properties of detuned, correlatedoscillators has been analyzed in Ref. [18]. An experimentalinvestigation of Gaussian discord was performed in Ref. [5].

It is natural to investigate CV quantum discord beyondGaussian measurements: non-Gaussian ones may indeed allowfor a stronger minimization of discord, and in this case theGaussian discord would be an overestimation of the truediscord. Here we focus on Gaussian states and ask whetherGaussian measurements are optimal in this case, i.e., whetherthe Gaussian discord is the true discord for Gaussian states.This question is relevant for two main reasons: First, ifdiscord is a truly useful resource for quantum informationprotocols [5,6], then it is crucial to have a reliable estimate ofits actual value. Second, from a fundamental point of view itis important to establish how different kinds of measurementscan affect correlations in quantum states. A further motivationcomes from the fact that indeed for some non-Gaussian states,e.g., CV Werner states, non-Gaussian measurements suchas photon counting have been proven to lead to a betterminimization [19].

The optimality of Gaussian measurements has already beenproven analytically for two-mode Gaussian states having onevacuum normal mode [15], by use of the so-called Koashi-Winter relation [20], but no analytic argument is available inthe general case. We address the question numerically, forthe case of two-modes, upon considering two large classesof Gaussian states, the squeezed thermal states (STS) andthe mixed thermal states (MTS), and allowing for a rangeof experimentally feasible non-Gaussian measurements basedon orthogonal bases: the number basis, the squeezed numberbasis, and the displaced number basis. As a result, we provideevidence that Gaussian quantum discord is indeed optimal for

052328-11050-2947/2012/86(5)/052328(12) ©2012 American Physical Society

PAOLO GIORDA, MICHELE ALLEGRA, AND MATTEO G. A. PARIS PHYSICAL REVIEW A 86, 052328 (2012)

the states under study. In addition, we also investigate the CVgeometric discord [21], comparing the case of Gaussian andnon-Gaussian measurements.

This work is structured as follows. In Sec. II we reviewquantum discord and the Gaussian version of it; in Sec. IIIwe thoroughly describe the basic question we want to addressin this work and introduce non-Gaussian measurements andnon-Gaussian discord; in Secs. IV–VI, we present our keyresults concerning non-Gaussian discord upon measurementsin the number basis, squeezed number basis, and displacednumber basis; in Sec. VII we discuss the behavior of non-Gaussian geometric discord; finally, Sec. VII closes the paperdiscussing our main conclusions.

II. QUANTUM DISCORD AND GAUSSIAN DISCORD

Starting from the seminal works by Ollivier and Zurek [2]and Henderson and Vedral [3], various measures of quantumcorrelations which go beyond the traditional entanglementpicture have been defined [4]. The most common measureof such correlations is the quantum discord [2,3]. Let usconsider a bipartite system composed of subsystems A andB. The total correlations in the global state are measured bythe mutual information I (A : B) = S(!A) + S(!B) − S(!AB).Whenever I (A : B) > 0, the subsystems are correlated andwe can gain some information about A by measurementson B only. However, there is no unique way of locallyprobing the state of B: to do it, we can perform differentlocal measurements or POVMs. Any such local POVM "B

is specified by a set of positive operators {"xB = Mx

BMx†B }

on subsystem B summing up to the identity∑

x "xB = I.

When measurement result x is obtained, the state of A isprojected onto !x

A = TrB[MxB!ABM

x†B ]. The uncertainty on

the state of A before the measurement on B is given byS(!A), while the average uncertainty on the state of A afterthe measurement is given by the average conditional entropyS"(A|B) =

∑x pxS(!x

A). Their difference

S(!A) − S"(A|B) = S(!A) −∑

x

pxS(!x

A

)

represents the average gain of information about the state ofA acquired through a local measurement on B. The maximalgain of information that can be obtained with a POVM,

C(A : B) = max{"∈POVM}

[S(!A) − S"(A|B)]

= S(!A) − min{"∈POVM}

[S"(A|B)], (1)

coincides with the measure of classical correlations originallyderived in [3] under some basic and natural requirementsfor such a measure. Quantum discord is then defined as thedifference between the mutual information and the classicalcorrelations:

D(A : B) = I (A : B) − C(A : B) (2)

and measures the part of the total correlations that cannot beexploited to gain information on A by a local measurement onB, i.e., measures the additional quantum correlations beyondthe classical ones. As discussed in [22], the minimization inEq. (1) is always achieved by rank-one POVMs.

It can be verified (see, e.g., [12]) that we have S(!A) −S"(A|B) = I"(A : B) = S(!"

A ) + S(!"B ) − S(!"

AB) for allrank-one POVMs, where the unconditional postmeasure-ment states are given by !"

AB =∑

x MxB!ABM

x†B , !"

A =TrB[

∑x Mx

B!ABMx†B ], !"

B = TrA[∑

x MxB!ABM

x†B ]. There-

fore, the classical correlations coincide with the mutualinformation in the system after the measurement, maximizedover all possible (rank-one) POVMs:

C(A : B) = max{"∈POVM}

I"(A : B), (3)

and the quantum discord coincides with the difference betweenthe mutual information before and after the measurement,minimized over all possible POVMs:

D(A : B) = min{"∈POVM}

[I (A : B) − I"(A : B)]. (4)

From the previous considerations, it is clear that D(A : B) = 0if and only if there is a local rank-one POVM "B which leavesthe global state of the system unaffected: ∃ ", !AB = !"

AB .Such states are called quantum-classical states and are in theform

χAB =∑

i

pi!A,i ⊗ |i〉〈i|, (5)

where pi is a probability distribution and {|i〉} is a basis forthe Hilbert space of subsystem B. For such states, there existsat least one local measurement that leaves the state invariantand we have I (A : B) = C(A : B), which means that we canobtain maximal information about subsystem A by a localmeasurement on B without altering the correlations with therest of the system.

In the realm of continuous-variable systems, the Gaussiandiscord [14,15] is defined by restricting the set of possiblemeasurements in Eq. (1) to the set of Gaussian POVMs [16],and minimizing only over this set. The Gaussian discord canbe analytically evaluated for two-mode Gaussian states, whereone mode is probed through (single-mode) Gaussian POVMs.The latter can be written in general as

"B(η) = π−1DB(η)!MD†B(η),

where DB(η) = exp(ηb† − η∗b) is the displacement operator,and !M is a single-mode Gaussian state with zero mean andcovariance matrix σM = ( α γ

γ β ). Two-mode Gaussian states can

be characterized by their covariance matrix σAB = ( A CCT B ). By

means of local unitaries that preserve the Gaussian characterof the state, i.e., local symplectic operations, σAB may bebrought to the so-called standard form, i.e., A = diag(a,a),B = diag(b,b), C = diag(c1,c2). The quantities I1 = det A,I2 = det B, I3 = det C, and I4 = det σAB are left unchangedby the transformations, and are thus referred to as symplecticinvariants. The local invariance of the discord has therefore twomain consequences: (i) correlation measures may be writtenin terms of symplectic invariants only, and (ii) we can restrictto states with σ already in the standard form. Before themeasurement we have

S(!AB) = h(d+) + h(d−), (6)

S(!A) = h(√

I1), S(!B) = h(√

I2), (7)

052328-2

QUANTUM DISCORD FOR GAUSSIAN STATES WITH NON- . . . PHYSICAL REVIEW A 86, 052328 (2012)

where h[x] = (x+1/2) ln(x+1/2)−(x − 1/2) ln(x − 1/2)and d± are the symplectic eigenvalues of !AB expressed byd2

± = 1/2[* ±√

*2 − 4I4], * = I1 + I2 + 2I3. After themeasurement, the (conditional) post-measurement state ofmode A is a Gaussian state with covariance matrix σP thatis independent of the measurement outcome and is givenby the Schur complement σP = A − C(B + σM )−1CT . TheGaussian discord is therefore expressed by

DG(A : B) = h(√

I2) − h(d−) − h(d+) + minσM

h(det√

σP ),

(8)

where we use two key properties: (i) the entropy of a Gaussianstate depends only on the covariance matrix, and (ii) thecovariance matrix σP of the conditional state does not dependon the outcome of the measurement. The minimization overσM can be done analytically. For the relevant case of stateswith C = diag(c,±c), including STS and MTS (see below),the minimum is obtained for α = β = 1/2 and γ = 0, i.e.,when the covariance matrix of the measurement is the identity.This corresponds to the coherent state POVM, i.e., to thejoint measurement of canonical operators, say position andmomentum, which may be realized on the radiation fieldby means of heterodyne detection. For separable states theGaussian discord grows with the total energy of the state andit is bounded, D ! 1; furthermore, we have D = 0 if and onlyif the Gaussian state is in product form !AB = !A ⊗ !B .

III. NON-GAUSSIAN DISCORD

In this work we consider Gaussian states, and ask whethernon-Gaussian measurements can allow for a better extractionof information than Gaussian ones, hence leading to lowervalues of discord. The optimality of Gaussian measurementshas been already proven for a special case [15], that oftwo-mode Gaussian states having one vacuum normal mode.Indeed any bipartite state !AB can be purified, !AB =⇒|ψ〉ABC ; then, the Koashi-Winter relation [20]

D(A : B) = Ef (A : C) + S(!B) − S(!AB) (9)

relates the quantum discord D and the entanglement offormation Ef of reduced states !AB and !AC respectively.Given a (mixed) two-mode Gaussian state !AB , there exists aGaussian purification |ψ〉ABC . In general, the purification of!AB requires two additional modes, so that !AC is a three-modeGaussian state. In the special case when one normal mode isthe vacuum, the purification requires one mode only. In thiscase, !AC represents a symmetric two-mode Gaussian state, aswas shown in [15]. Hence, Ef (A : C) can be evaluated fromRef. [23] where an analytical formula for the entanglement offormation of symmetric two-mode Gaussian states is given.From Ef (A : C), by means of Eq. (9) of Ref. [20], one canobtain D(A : B) (the exact discord) and a comparison withDG(A : B) proves that D(A : B) = DG(A : B).

In the general case, there is no straightforward analyticalway to prove that Gaussian discord is optimal. Therefore, weperform a numerical study. Since taking into account the mostgeneral set of non-Gaussian measurements is an extremelychallenging task, one can rather focus on a restricted subset. Wechoose to focus on a class of measurements that are realizable

with current or foreseeable quantum optical technology. Theseare the the projective POVMs, " = {"n}, represented by thefollowing orthogonal measurement bases:

"n = D(α)S(r)|n〉〈n|S(r)†D(α)†, n = 0, . . . ,∞, (10)

where S(r) = exp (−r∗ a2

2 − r (a†)2

2 ) and D(α) = exp(αa† −α∗a) are, respectively, the single-mode squeezing and dis-placement operators [24]. The set of projectors in Eq. (10)is a POVM for any fixed value of α and r . If α = r = 0 wehave the spectral measure of the number operator, describingideal photon counting "n = |n〉〈n|. If α > 0,r = 0 we areprojecting onto displaced number states [25], if α = 0,r > 0onto squeezed number states [26–29]. The class (10) encom-passes some of the most relevant measurements that can berealistically accessed in experiments [30].

In the following, we will evaluate the non-Gaussianquantum discord defined by

DNG(A : B) = h(√

I2) − h(d−) − h(d+) + S",NG(A|B),

(11)

where the non-Gaussian measurements are given by Eq. (10)above. For the non-Gaussian conditional entropy we have

S",NG(A|B) =∑

n

pnS(!A,n) ,

!A,n = 1pn

TrB["n!AB"n] , (12)

pn = TrAB["n!AB"n].

In the following we consider two classes of Gaussian states inorder to assess the performances of the above measurements.These are the two-mode squeezed thermal states (STS) [31–33]

! = S(λ)ν1(N1) ⊗ ν1(N2)S(λ)†, (13)

and the two-mode mixed thermal states (MTS) [34]

ρ = U (φ)ν1(N1) ⊗ ν1(N2)U (φ)†, (14)

where νi(Ni) are one-mode thermal states with thermal photonnumber Ni ; S(λ) = exp{λ(a†

1a†2 − a1a2)} is the two-mode

squeezing operator (usually realized on optical modes throughparametric down-conversion in a nonlinear crystal); andU (φ) = exp{φ(a†

1a2 − a1a†2)} is the two-mode mixing operator

(usually realized on optical modes through a beam splitter).Notice that the STS and MTS correspond respectively toc1 = −c2 and c1 = c2 in the standard form of the covariancematrix given in Sec. II. Furthermore, the MTS are alwaysseparable since mixing two thermal states cannot createentanglement [35].

In particular, in the following we will focus on the simplestcase of symmetric STS with N1 = N2 ∈ [10−5,1] λ ∈ [0,0.5].As for MTS, we cannot consider the symmetric case (sinceif N1 = N2 then the mutual information vanishes and thereare no correlations in the system), therefore we considerthe unbalanced case and focus on φ ∈ [0,π/2] and N1,N2 ∈[10−5,1].

052328-3

PAOLO GIORDA, MICHELE ALLEGRA, AND MATTEO G. A. PARIS PHYSICAL REVIEW A 86, 052328 (2012)

IV. NUMBER BASIS

Let "n = |n〉〈n|. In this case, the postmeasurement state is

!An ⊗ |n〉〈n| =

(∑

h,k

!(h,k),(n,n)|h〉〈k|)

⊗ |n〉〈n|, (15)

and we have the following expression for the density matrixelements

!(h,k),(n,n) =∑

s,t

pths (N1)pth

t (N2)Ohn(st)O∗kn(st), (16)

where pths (N ) = Ns(1 + N )−(s+1) and Ohn(st) = 〈hn|O|st〉

with O = S(λ),U (φ) for STS and MTS, respectively. Thepostmeasurement state !A

n is diagonal (see Appendix A),

〈h|!An |k〉 = δhk !(h,h),(n,n). (17)

As a consequence, the entropy of the postmeasure-ment state can be expressed as S(!A

n ) = H ({!(h,h),(n,n)}) =H ( +p(A|B = n)), where H is the Shannon entropy of the con-ditional probability +p(A|B = n) = (p(0,n),p(1,n), · · · )/pn,and therefore the overall conditional entropy can be simplyexpressed in terms of the photon number statistics:

S(A|{"n}) =∑

n

pnH ( +p(A|B = n))

= H ( +p(A,B)) − H ( +p(B)), (18)

with +p(A,B) = {p(A = n,B = m)} and +p(B) = {p(B = n)}.In view of this relation, the only elements of the number basisrepresentation of the density matrix ! that are needed are thediagonal ones, i.e., one has to determine the photon numberstatistics for the two-mode STS or MTS. The required matrixelements can be obtained in terms of the elements of the two-mode squeezing and mixing operators (see Appendix A). Onehas of course to define a cutoff on the dimension of the densitymatrix. This can be done upon requiring that the error onthe trace of each state considered be sufficiently small: 1 −Tr!trunc < εerr. This implies that the truncated density matrix!trunc is very close to the actual one (notice that Tr[! − !trunc] <εerr implies ||! − !trunc||1 < 3

√εerr [36]).

We have compared Gaussian and non-Gaussian quantumdiscord (with the non-Gaussian measurements correspondingto photon number measurements) for STS and MTS with awide range of squeezing, mixing and thermal parameters.In Fig. 1 we show results for STS with varying λ andN1 = N2 = 10−2, N1 = N2 = 1. The key result is that thenon-Gaussian quantum discord is always greater than itsGaussian counterpart for all values of N1 and λ. The gapgrows with increasing N1 and λ. In Fig. 2 we show results forMTS N1 = {0.1,1} and q = N2/N1 = {0,0.1,0.4,0.5}. Alsoin this case, the non-Gaussian discord is always higher thanthe Gaussian one.

Both results indicate that the Gaussian (heterodyne) mea-surement is optimal for STS and MTS, at least comparedto photon counting, in the sense that it allows for a betterextraction of information on mode A by a measurement onmode B.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.1 0.2 0.3 0.4 0.5

G, N1 = N2 = 10−2

G, N1 = N2 = 1nG, N1 = N2 = 10−2

nG, N1 = N2 = 1

FIG. 1. (Color online) Gaussian and non-Gaussian quantumdiscord for STS as a function of λ, for different values of N1 = N2.

V. SQUEEZED NUMBER BASIS

We now analyze the case of non-Gaussian measure-ments represented by the squeezed number basis |nr〉〈nr | =S(r)|n〉〈n|S(r)†, where S(r) = exp (−r∗ a2

2 − r (a†)2

2 ) is thesingle-mode squeezing operator. A local measurement in thesqueezed number basis is equivalent to a measurement inthe number basis, performed on a locally squeezed state. In

0

0.02

0.04

0.06

0.08

0.1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

G, Ν1 = 0.1, θ = 0nG, Ν1 = 0.1, θ = 0G, Ν1 = 0.1, θ = 0.5

nG, Ν1 = 0.1, θ = 0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1 1.2 1.4

G, Ν1 = 1, θ = 0nG, Ν1 = 1, θ = 0

G, Ν1 = 1, θ = 0.5nG, Ν1 = 1, θ = 0.5

FIG. 2. (Color online) Gaussian and non-Gaussian quantumdiscord for MTS as a function of φ for different values of N1 andq = N2/N1.

052328-4

QUANTUM DISCORD FOR GAUSSIAN STATES WITH NON- . . . PHYSICAL REVIEW A 86, 052328 (2012)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.1 0.2 0.3 0.4 0.5

GnG, r = 0

nG, r = 0.5

FIG. 3. (Color online) Gaussian and non-Gaussian quantumdiscord for STS with N1 = 1 as a function of λ and for differentvalues of local squeezing r .

formulas, the probability of measuring nr on one subsystemfor state ! is given by

p!(nr ) = Tr(1 ⊗ |nr〉〈nr |!) = Tr[1 ⊗ |n〉〈n|S†(r)!S(r)]

= Tr(1 ⊗ |n〉〈n|!r ) = p!r(n), (19)

i.e., it is equal to the probability of measuring n on the locallysqueezed state !r = S(r)!S(r)†, and the relative postmeasure-ment state is

!Anr

= TrB[1 ⊗ |nr〉〈nr |!1 ⊗ |nr〉〈nr |]/p!(nr )

= TrB[1 ⊗ |n〉〈n|!r1 ⊗ |n〉〈n|]/pr!(n) = !A

rn. (20)

The general idea is that measurements on a state ! in a basis thatis obtained by performing a unitary (Gaussian) operation V onthe number basis |n〉〈n| can be represented as measurementson the number basis of a modified state !V = V !V † on whichthe local unitary operation acts.

In the case of the squeezed number basis, the postmea-surement state is not diagonal, therefore the reasoning leadingto Eq. (18) does not hold. The postmeasurement state matrixelements (!A

rn)h,k = !(h,k),(n,n) can be obtained directly by eval-

uating the expression (16) where now the expression Ohk(st) =〈hk|O|st〉 [where O = S(λ),U (φ)] must be substituted withO ′

hk(st) = 〈hk|S(r)O|st〉 =∑

q〈k|S(r)|q〉〈hq|O|st〉, and theelements of the single-mode squeezing operator are givenin [37] (Eq. 20) or in [27] (Eq. 5.1).

We have evaluated the Gaussian and non-Gaussian quantumdiscord for STS and MTS with a wide range of two-modesqueezing and thermal parameters. Non-Gaussian measure-ments are done in the squeezed photon number basis, "n =S(r)|n〉〈n|S(r)† with variable r ∈ [0,0.5]. The effect of localsqueezing on non-Gaussian quantum discord is negligible inthe whole parameter range under consideration: we comparethe non-Gaussian discord for different values of r and findthat all curves collapse. This can be seen in Figs. 3 and 4,which plot the behavior for N1 = N2 = 0.01 (STS) andN1 = 1,N2 = 0 (MTS). The same behavior is observed in thewhole parameter range under investigation. We have verifiednumerically that the postmeasurement states of mode A!A

rn

are not equal as r varies (i.e., the postmeasurement statescorresponding to measurement result nr change with r), yet

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1 1.2 1.4

GnG, r = 0

nG, r = 0.5

FIG. 4. (Color online) Gaussian and non-Gaussian quantumdiscord for MTS for N1 = 1, N2 = 0 as a function of φ and fordifferent values of local squeezing r .

the sum∑

n pnS(!Arn

) is equal for all values of r underinvestigation. Therefore, the squeezing in the measurementbasis has no effect on the discord, at least for the values ofsqueezing considered: in particular, it cannot afford a deeperminimization than that obtained without local squeezing. Thisindicates that the heterodyne measurement remains optimalalso with respect to measurement in the squeezed numberbasis.

VI. DISPLACED NUMBER BASIS

We finally analyze the case of non-Gaussian measure-ments represented by the displaced number basis |nα〉〈nα| =D(α)|n〉〈n|D(α)†, where D(α) = exp(αa† − α∗a) is thesingle-mode displacement operator. According to the generalconsiderations above, a local measurement in the displacednumber basis is equivalent to a measurement in the numberbasis, performed on a locally displaced state !α . As in thecase of the squeezed number basis, the postmeasurement stateis not diagonal and we need all matrix elements (!A

αn)h,k =

!(h,k),(n,n). They can be obtained directly by evaluating theexpression (16) where the expression Ohk(st) = 〈hk|S(λ)|st〉[where O = S(λ),U (φ)] must be substituted with O ′

hk(st) =〈hk|D(α)O|st〉 =

∑q〈k|D(α)|q〉〈hq|O|st〉, and the elements

of the single-mode displacement operator are given in [38](Eq. 1.46).

The evaluation of the non-Gaussian quantum discord canbe simplified by first noticing that one can consider real valuesof α only. Indeed, the quantum discord only depends onthe modulus |α|. This is shown in detail in Appendix B byusing the characteristic function formalism. Consider !A

nα, the

postmeasurement state of mode A after measurement result nα

is obtained on B. If we change the phase of α, α → α′ ≡ eiθα,we find that

!Anα′ = U!A

nαU †, (21)

where U is a unitary operation corresponding to a simplequadrature rotation

a1 → a1eiθ , a

†1 → a

†1e

−iθ . (22)

052328-5

PAOLO GIORDA, MICHELE ALLEGRA, AND MATTEO G. A. PARIS PHYSICAL REVIEW A 86, 052328 (2012)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.1 0.2 0.3 0.4 0.5

GnG, α = 0nG, α = 1.0

FIG. 5. (Color online) Gaussian and non-Gaussian quantumdiscord for STS with N1 = 1 as a function of λ and for differentvalues of local displacement α.

Therefore, we have !Aα′

n/= !A

αn, but !A

α′n

and !Aαn

have the samespectrum, since they are related by a unitary. Therefore, theentropy of the reduced postmeasurement state !A

α does notdepend on the phase of α but just on |α|. If follows that thenon-Gaussian quantum discord of !α does not depend on thephase of α.

We have evaluated the Gaussian and non-Gaussian quantumdiscord for STS and MTS with a wide range of two-modesqueezing and thermal parameters. Non-Gaussian measure-ments are done in the displaced photon number basis, "n =D(α)|n〉〈n|D(α)† with variable α ∈ [0,2.5]. In Figs. 5 and 6we plot the Gaussian and non-Gaussian quantum discord. Wesee that greater displacements lead to lower values of the non-Gaussian quantum discord, but the decrease is insufficient tomatch the Gaussian quantum discord, which remains optimal.However, the non-Gaussian quantum discord approximates theGaussian one as α → ∞. This is analytically proven below inAppendix C. There we find that for both STS and MTS,

!Aαn

→ !Aα0

as α → ∞, (23)

i.e, the conditional states !Aαn

becomes independent of n andequal to the n = 0 result. As a consequence, the conditionalentropy in the displaced number basis is equal to the entropyof the postmeasurement state for any measurement result, and,in particular, for n = 0:

S",NG(A|B) =∑

n

pnS(!A

αn

)→ S

(!A

α0

)as α → ∞.

(24)

But !Aα0

is just the postmeasurement state we obtain after aheterodyne detection on mode B (equal for all measurementresult modulo a phase space translation which is irrelevant asfor the entropy). Therefore, we also have SG(A|B) = S(!A

α0)

and the non-Gaussian discord DNG(A : B) in the displacednumber basis tends to the Gaussian discord DG(A : B) asα → ∞.

Actually, we cannot prove that the DNG(A : B) is lowerbounded by DG(A : B), and we cannot rule out the possibilitythat DNG(A : B) < DG(A : B) for intermediate values of α.However, our numerical data do not support this possibility

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1 1.2 1.4

GnG, α = 0

nG, α = 1.0

FIG. 6. (Color online) Gaussian and non-Gaussian quantumdiscord for MTS for N1 = 1, N2 = 0 as a function of φ and fordifferent values of local displacement α.

since we never observe DNG(A : B) < DG(A : B) and weexpect that DNG(A : B) → DG(A : B) from above as α → ∞.In conclusion, we have analytical and numerical evidence thatthe heterodyne measurement remains optimal also with respectto measurement in the displaced number basis.

VII. GEOMETRIC DISCORD

In this section, we briefly consider the recently introducedmeasure of geometric discord and compare results with thoseobtained for the quantum discord.

A. Definition and properties

Geometric discord [12] is defined as

DG(A : B) = min{χAB∈C}

‖!AB − χAB‖2, (25)

and it measures the distance of a state from the set Cof quantum-classical states where ‖A‖2 = Tr[A†A] is theHilbert-Schmidt distance. Clearly DG = 0 if and only ifD = 0, since both measures vanish on the set of classicallycorrelated states. In particular, it has been be proven that DG

can be seen a measure of the discrepancy between a statebefore and after a local measurement on subsystem B [39]:

DG(A : B) = min{"∈POVM}

∥∥!AB − !"AB

∥∥2, (26)

where the unconditional postmeasurement state is givenby !"

AB =∑

x MxB!ABM

x†B . Notice that DG and D are not

monotonic functions of one another and the relation betweenthem is still an open question. However, in many cases DG ismuch simpler to evaluate than D.

Analogous to the case of Gaussian discord, a Gaussianversion of the geometric discord can be defined by restrictingto Gaussian measurements [21]. Again, it can be analyticallycomputed for two-mode Gaussian states. With the samereasoning used in Sec. II one easily obtains

DGG(!AB) = min

σM

Tr[(!AB − !P ⊗ !M )2]. (27)

052328-6

QUANTUM DISCORD FOR GAUSSIAN STATES WITH NON- . . . PHYSICAL REVIEW A 86, 052328 (2012)

Exploiting the property that Tr[!1!2] = 1/ det[(σ1 + σ2)/2],for any two Gaussian states !1 and !2,

DG(A : B) = minσM

{1/√

det σAB + 1/√

det(σP ⊕ σM )

− 2/√

det[(σAB + σP ⊕ σM )/2]}. (28)

For for the relevant case of STS and MTS, the mini-mum is obtained with the σM elements given by α =β =

√ab(

√4ab−3c2+

√ab)

3a, γ = 0. The least disturbing Gaussian

POVM for STS, according to the Hilbert-Schmidt distance,is thus a (noisy) heterodyne detection, a result which isanalogous to what is found in the case of quantum discord.If one constrains the mean energy per mode, the Gaussianquantum discord gives upper and lower bounds to theGaussian geometric discord. In absence of such a constraint,the geometric discord can vanish for arbitrarily stronglynonclassical (entangled) Gaussian states; for instance, asreported in Ref. [21], STSs with b = 1 + ε, c =

√(a + 1)ε

with 0 ! ε ! a − 1 have diverging discord and vanishinggeometric discord for a → ∞.

Also in this case, we may consider non-Gaussian measure-ments and evaluate a non-Gaussian geometric discord:

DNGG (A : B) = Tr

[(!AB − !"

AB

)2]. (29)

For measurement in the number basis, we can easily obtain

DNGG = µ(!) +

∑

npq

|〈pn|!|qn〉|2, (30)

where µ(!) = 14√

det(σ ) is the purity of the Gaussian state [38].In the case of measurements in the squeezed or displacednumber basis, we have to use !r and !α instead of ! inEq. (30). In general, to compute the geometric discord, we needto compute matrix elements, and we use the same numericalmethods described above.

B. Results

We have compared the Gaussian and non-Gaussian geomet-ric discord for STS and MTS in a wide range of parameters.We have considered measurements in the number, squeezednumber, and displaced number basis for the same values of theparameters given in the preceding sections. Results are plottedin Figs. 7 and 8. In general, at variance with the results forquantum discord, we find that non-Gaussian measurementscan provide lower values of geometric discord than Gaussianones. Among the class of non-Gaussian measurements we haveconsidered, the optimal one is provided by the number basis,which gives values of geometric discord that are always lowerthan those given by the optimal Gaussian measurement. Thenon-Gaussian geometric discord increases for increasing r andα, and it can become greater than its Gaussian counterpart.These results are very different from the quantum discordcase: on one hand, the (non-Gaussian) geometric discord issubstantially affected by the local squeezing; on the other hand,it does not approach the Gaussian one when the displacementα → ∞, but it grows monotonically. Indeed if we increasethe squeezing or displacement in the measurement basis, thepostmeasurement state is more distant (in Hilbert-Schmidtnorm) from the original one. As already noticed, performingthe measurement in the squeezed (displaced) number basis

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.2 0.4 0.6 0.8 1 1.2 1.4

GnG, r = 0

nG, r = 0.5

0

0.01

0.02

0.03

0.04

0.05

0 0.1 0.2 0.3 0.4 0.5

GnG, r = 0

nG, r = 0.5

FIG. 7. (Color online) Gaussian and non-Gaussian geometricdiscord for STS with N1 = 1 as a function of λ and for different valuesof local squeezing r (top); Gaussian and non-Gaussian geometricdiscord for MTS for N1 = 1, N2 = 0 as a function of φ and fordifferent values of local squeezing r (bottom).

in equivalent to first squeezing (displacing) the state andthen measuring it in the number basis. The local squeezingand displacement have the effect of increasing the energy ofthe state, shifting the photon number distribution P (B = n)towards greater values of n. This causes the overlap betweenthe postmeasurement state and the original state to decrease,and therefore their distance to increase.

Let us further comment on the difference between thequantum discord and the geometric discord cases. Quantumdiscord and geometric discord both vanish for classical states,but are not monotonic functions of one another, and thusthey are truly different quantities. Geometric discord, basedon the Hilbert-Schmidt distance, is a geometric measure ofhow much a state is perturbed by a local measurement, whilequantum discord assesses to which extent correlations areaffected by a local measurement. While for quantum discordwell-defined operational and informational interpretations canbe found [5,6,40], for geometric discord the situation is moreproblematic. Indeed, one can design protocols in which thegeometric discord can in some cases be related to the protocols’performances [8,41]; however, recent discussions [42], showthat as consequence of the noninvariance of the Hilbert-Schmidt norm under quantum evolutions, it is difficult to find aconclusive argument about the relevance of geometric discord

052328-7

PAOLO GIORDA, MICHELE ALLEGRA, AND MATTEO G. A. PARIS PHYSICAL REVIEW A 86, 052328 (2012)

0

0.02

0.04

0.06

0.08

0.1

0 0.1 0.2 0.3 0.4 0.5

GnG, α = 0nG, α = 1.0

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.2 0.4 0.6 0.8 1 1.2 1.4

GnG, α = 0

nG, α = 1.0

FIG. 8. (Color online) Gaussian and non-Gaussian geometricdiscord for STS with N1 = 1 as a function of λ and for differentvalues of local displacement α (top); Gaussian and non-Gaussiangeometric discord for MTS for N1 = 1, N2 = 0 as a function of φ

and for different values of local displacement α (bottom).

as a measure of quantumness of correlations. Our data showthat non-Gaussian measurements can yield optimal values ofthe geometric discord, contrary to the case of quantum discord.Hence, the behavior of quantum discord and geometric discordwith respect to different types of measurements is different.This is a further indication that geometric discord cannot beused as a good benchmark for quantum discord and that thedegree of quantumness measured, if any, by such a quantityhas a fundamentally different nature.

VIII. DISCUSSION AND CONCLUSIONS

The definition of discord involves an optimization overall possible local measurements (POVMs) on one of thesubsystems of a bipartite composite quantum system. In therealm of continuous variables (CVs), initial research effortson quantum discord restricted the minimization to the set of(one-mode) Gaussian measurements.

In this work we have investigated CV quantum discordbeyond this restriction. We have focused on Gaussian states,asking whether Gaussian measurements are optimal in thiscase, i.e., whether Gaussian discord is the true discord forGaussian states. While a positive answer to this question had

already been given for the special case of two-mode Gaussianstates having one vacuum normal mode (by means of ananalytical argument based on the Koashi-Winter formula),no general result was available so far. We have addressedour central question upon considering two large classes oftwo-mode Gaussian states, the squeezed thermal states (STS)and the mixed thermal states (MTS), and allowing for a widerange of experimentally feasible non-Gaussian measurementsbased on orthogonal bases: the photon number basis, squeezednumber basis, and displaced number basis. For both STS andMTS, in the range of parameters considered, the Gaussianmeasurements always provide optimal values of discordcompared to the non-Gaussian measurements under analysis.Local squeezing of the measurement basis has no appreciableeffect on correlations, while local displacement leads to lowervalues of the non-Gaussian discord, which approaches theGaussian one in the limit of infinite displacement.

Overall, for the explored range of states and measurements,we have evidence that Gaussian discord is the ultimate quan-tum discord for Gaussian states. We note that the optimalityof Gaussian measurements suggested by our analysis is aproperty which holds only for Gaussian states. In the caseof non-Gaussian states, e.g., CV Werner states, non-Gaussianmeasurements such as photon counting can lead to a betterminimization, as was recently proven in Ref. [19].

We also have investigated the CV geometric discord [21],comparing the Gaussian and non-Gaussian cases. We haveshown that the behavior of geometric discord is completelydifferent from that of quantum discord. On the one hand,non-Gaussian measurements can lead to lower values of thegeometric discord, the number basis measurement being theoptimal one; on the other hand, the effects of both localsqueezing and displacement are strong and consist in anoteworthy increase in the non-Gaussian geometric discord.The remarkable differences between quantum and geometricdiscord imply that the latter cannot be used as a benchmark ofthe former.

Both in the case of the discord and geometric discord adefinite answer on the optimal measurement minimizing therespective formulas would require the extension of the setof non-Gaussian measurements to possibly more exotic onesand the application of those realizable in actual experimentsto a broader class of Gaussian and non-Gaussian states.While we leave this task for future research, our results ondiscord support the conjecture that Gaussian measurementsare optimal for Gaussian states and allow us to set, forthe class of states analyzed, a tighter upper bound on theentanglement of formation for 1 × 2 mode Gaussian states,via the Koashi-Winter relation.

APPENDIX A: POSTMEASUREMENT STATEIS DIAGONAL

We prove that the postmeasurement state

!An = TrB[1 ⊗ |n〉〈n| ! 1 ⊗ |n〉〈n|]/pn (A1)

of STS and MTS after local measurement in the numberbasis is diagonal [here, pn = Tr(!1 ⊗ |n〉〈n|)]. We have

052328-8

QUANTUM DISCORD FOR GAUSSIAN STATES WITH NON- . . . PHYSICAL REVIEW A 86, 052328 (2012)

indeed

! =∑

s,t

pths (N1)pth

t (N2)O|st〉〈st |O†

=∑

(h,n),(k,m)

|hn〉〈km|(

∑

s,t

pths pth

t Ohn(st)O∗km(st)

)

, (A2)

where pths (N ) = Ns (1 + N )−(s+1), where Ohn(st) =

〈hn|O|st〉 and O∗km(st) = 〈st |O†|km〉 = 〈km|O|st〉∗, and

where O = S(λ), O = U (φ) for STS and MTS, respectively.The postmeasurement states can be written as

!An ⊗ |n〉〈n| =

(∑

h,k

!(h,k),(n,n)|h〉〈k|)

⊗ |n〉〈n|, (A3)

and therefore we need to evaluate the matrix elements

!(h,k),(n,n) =∑

s,t

pths pth

t Ohn(st)O∗kn(st). (A4)

The elements of the two-mode squeezing operator are givenin [29] [Eq. (22)]:

〈hn|S(λ)|st〉= δt+h,s+nf

λ(h,n,s,t) = δt+h,s+n

×min(s,t)∑

a=0

min(h,n)∑

b=0

(−1)a+b(sech λ)t+h−a−b−1µa−b+h−s

× (t + h − a − b)![s!t!h!n!]1/2

a!(t − a)!(s − a)!b!(n − b)!(h − b)!, (A5)

where µ = eλ, while the elements of the two-mode mixingoperator

〈hn|U (φ)|st〉 = δh+n,s+t

min{h,s}∑

a=max{0,h−t}Ast

a h−a

= δh+n,s+t

min{h,s}∑

a=max{0,h−t}

√h!(s + t − h)!

s!t!(−1)h−a

×(

s

a

)(t

h − a

)sin φs+h−2a cos φt+2a−h.

(A6)

To evaluate Eq. (A4), we need Ohn(st)O∗kn(st). Due to the δ’s

appearing in both Eqs. (A5) and (A6), the following relationsmust be satisfied:

t − s = n − h, t − s = n − k for O = S(λ), (A7)

t + s = n + h, t + s = n + k for O = U (φ), (A8)

and both equations imply h = k; therefore the postmeasure-ment state is diagonal in the number basis:

(!A

n

)h,k

= δh,k

∑

s,t

pths (N1)pth

t (N2)|Ohk(st)|2. (A9)

APPENDIX B: DISCORD DOES NOT DEPENDON THE PHASE OF DISPLACEMENT

We show that the (non-Gaussian) discord in the displacednumber basis does not depend on the phase of displace-

ment for STS and MTS. The argument is best given inthe characteristic function representation of the states [38],χ [!](λ) = Tr[!D(λ)]. The STS and MTS have a Gaussiancharacteristic function χ [!](!) = exp(− 1

2!T σ!) where ! =1√2( Re λA, Im λA, Re λB, Im λB) and the covariance matrix is

given by

σ =(

A C

CT B

)=

a 0 c 00 a 0 ±c

c 0 b 00 ±c 0 b

, (B1)

where ±c is −c in the case of STS and +c in the case of MTS.For STS we have

χ [!](λA,λB) = exp[−a|λA|2 − b|λB |2 + 2c Re (λAλB)],

(B2)

while for MTS the same expression holds upon changing2c Re (λAλB) → 2c Re (λ∗

AλB). In the following, we shallcarry on the argument for STS, but the MTS case is fullyequivalent. If we perform a displacement on one mode, ! →D(α)!D†(α) ≡ !α , the effect on the characteristic functionis easy to evaluate. Using the relation D(α)D(λ)D†(α) =D(λ) exp[−2i Im (λα∗)] [38] we obtain

χ [!α](λA,λB) = χ [!](λA,λB) exp[−2i Im (λBα∗)]. (B3)

Suppose we perform a measurement on mode B in the numberbasis {"n = |n〉〈n|}. The postmeasurement state of mode Ais !A

α,n = 1pn

TrB[!α"n] where pn = Tr[!α"n]. By use of thetrace formula [38]

Tr[O1O2] = 1π

∫

Cm

d2mλ χ [O1](λ)χ [O2](−λ),

we obtain the characteristic function

χ[!A

αn

](λA) = 1

πpn

∫

Cd2λB χ ["n](λB) χ [!α](λA,−λB).

(B4)

Since χ ["n](λB) = e− 12 |λB |2Ln(|λB |2), where Ln is the

Laguerre polynomial Ln(|λB |2) =∑n

i=0

(n

n−i

) |λB |2i

i! , we haveexplicitly

χ[!A

αn

](λA) = 1

πpn

∫

Cd2λB Ln(|λB |2)

× exp[−a|λA|2 − (b + 1/2)|λB |2

− 2c Re (λAλB) − 2i Im (λBα∗)]. (B5)

To see that this expression depends on |α| only, we canimplement the change α → α′ ≡ eiθα and we have

χ[!A

α′n

](λA) = 1

πpn

∫

Cd2λB Ln(|λB |2)

× exp[−a|λA|2 − (b + 1/2)|λB |2

− 2c Re (λAλB) − 2i Im (λBα∗e−iθ )].

By changing variable λB → e−iθλB we see that

χ[!A

α′n

](λA) = χ

[!A

αn

](λAe−iθ ). (B6)

052328-9

PAOLO GIORDA, MICHELE ALLEGRA, AND MATTEO G. A. PARIS PHYSICAL REVIEW A 86, 052328 (2012)

Therefore, we have χ [!Aα′

n] /= χ [!A

αn], hence !A

α′n/= !A

αn. How-

ever, !Aα′

nand !A

αnhave the same spectrum. Indeed χ [!A

αn(λA)]

and χ [!Aαn

(λAeiθ )] are related by a simple quadrature rotation

a1 → a1eiθ , a

†1 → a

†1e

−iθ , (B7)

which means that

!Aα′

n= U!A

αnU †, (B8)

where U is the free evolution of mode A, U = eiθa†1a1 . Since !A

α′n

and !Aαn

are related by a unitary, they have the same spectrum.Therefore, the spectrum (hence the entropy) of the reducedpostmeasurement state !A

α does not depend on the phase ofα but just on |α|. It follows that the non-Gaussian quantumdiscord of !α does not depend on the phase of α, Q.E.D.

As for the non-Gaussian geometric discord, it is obtainedas

Tr[(!)2] −∑

n

Tr[!A

αn!A

αn⊗ "B

n

]= Tr[(!)2] −

∑

n

Tr[!A

αn!A

αn

].

By the same arguments as used before, leading to Eq. (B8),we immediately see that the second trace does not depend onthe phase of α, hence the geometric discord does not either.

APPENDIX C: UNDERSTANDING THE BEHAVIORFOR GROWING α

Let us now consider in detail the behavior for growing α.We will show that the non-Gaussian discord in the displacednumber basis tends to the Gaussian discord as the displacementtends to infinity, DNG → DG as α → ∞. First, we will showthat

!Aαn

→ !Aα0

as α → ∞. (C1)

This is best shown in the characteristic function formalism.The postmeasurement state of mode A has the characteristicfunction (B5). Since the phase of α is irrelevant for the discord,we will assume α ∈ R in the following. The postmeasurementstate characteristic function, Eq. (B5), is the Gaussian integralof a polynomial. By using a well-known trick of Gaussianintegrals, we can rewrite

χ[!A

αn

](λA) = 1

πpn

e−a|λA|2∫

Cd2λB Ln(d/dγ )

× exp{−γ |λB |2 + 2c Re λA Re λB

− [2c Im (λA) + 2iα] Im (λB)},

evaluated at γ = b + 1/2 and the formal expression Ln(d/dγ )means

∑ni=0

(n

n−i

) 1i!

dn

dγ n . This expression can now be movedoutside the integral, so that we are now left with a purelyGaussian integral of the form

∫

Rd2!B exp

(−1

2!T

BM!B + !TBB

),

where M= diag{4γ ,4γ }, B= (2c Re λA,−2c Im λA+2iα), !B = (Re λB,Im λB). The integral gives

2π√detM

exp( 12B

TM−1B), so we finally get

χ[!A

αn

](λA) = 1

pn

exp[−1

2a|λA|2

]Ln(−d/dγ )

× 1γ

exp(c2|λA|2 − α2 − 2iαc Im λA

2γ

). (C2)

Let us define x = c2|λA|2 − α2 − 2iαc Im λA. Then we have

Ln(−d/dγ )1γ

exp(

c2|λA|2 − α2 − 2iαc Im λA

2γ

)

= Ln(−d/dγ )1γ

ex/2γ = Fn(γ ,x)ex/2γ ,

1 2 3 4 5

0.5

0.5

1.0

(a)α = 0.1

1 2 3 4 5

0.5

0.5

1.0

(b)α = 1

1 2 3 4 5

0.5

0.5

1.0

(c)α = 5

FIG. 9. (Color online) We show χ [!αAn

](λ): Real part (fullsymbols, green) and imaginary part (empty symbols, black), for n = 0(circles), n = 1 (squares), n = 2 (triangles), and different values of α.Here we have A = (NT + 1/2) cosh λ, C = (NT + 1/2) sinh λ, withλ = 0.5, NT = 0.5.

052328-10

QUANTUM DISCORD FOR GAUSSIAN STATES WITH NON- . . . PHYSICAL REVIEW A 86, 052328 (2012)

where Fn(γ ,x) =∑

k fk(γ )xk is necessarily a polynomial ofdegree n in with γ -dependent coefficients fk(γ ). Therefore,

χ[!A

αn

](λA) = 1

pn

exp{−[a − c2(b + 1/2)−1]|λA|2}

× exp[−ic(b + 1/2)−1α Im λA]

× exp(−α2/2γ )Fn(γ ,x). (C3)

The norm is

pn = exp{−[a − c2(b + 1/2)−1]|λA|2}

× exp[−ic(b + 1/2)−1α Im λA]

× exp(−α2/2γ )Fn(γ ,x)|λA=0

= exp(−α2/2γ )Fn(−α2),

so that

χ[!A

αn

](λA) = exp{−[a − c2(b + 1/2)−1]|λA|2}

× exp[−ic(b + 1/2)−1α Im λA]Fn(γ ,x)

Fn(γ , − α2).

(C4)

This function is exponentially decaying as e−s|λA|2 wheres = a − c2/(b + 1/2), hence it is vanishing for |λA|2 3 1/s.Therefore, we can consider values of |λA|2 in the regionλ2

A " 1/s. In this region, we have limα→∞ x = −α2 becauseα 3 λA and thus

limα→∞

Fn(γ ,x)Fn(γ , − α2)

= fn(γ )α2n

fn(γ )α2n= 1.

In conclusion, as α → ∞ we have

χ[!A

αn

](λA) → χ

[!A

α0

], (C5)

which implies the desired result (C1), Q.E.D.This result means that the conditional state of A is

independent of n and equal to the n = 0 result. In Fig. 9 weshow χ [!A

α0],χ [!A

α1],χ [!A

α2] for growing values of α. The three

curves converge already for α ∼ 5. As a consequence of nindependence, we have

S",NG(A|B) =∑

n

pnS(!A

αn

)→ S

(!A

α0

). (C6)

But !Aα0

is just the postmeasurement state corresponding to thePOVM element D(α)|0〉〈0|D†(α) = |α〉〈α|, i.e, a Gaussianstate with covariance matrix σP = A − C(B + I/2)−1CT

(Schur complement), and mean µP = X(B + I/2)−1CT ,where X = (α,0). On the other hand, from the discussionin Sec. II we know that the optimal Gaussian POVM is aheterodyne measurement {"β = D(β)|0〉〈0|D†(β) = |β〉〈β|}.In this case, as already explained in Sec. II, the entropyof the postmeasurement state !A

β is independent of themeasurement result β, hence the conditional entropy coincideswith the entropy of the β = α result. Therefore, we alsohave SG(A|B) = S(!α0 ). Therefore, we conclude that thenon-Gaussian discord DNG(A : B) in the displaced numberbasis tends to the Gaussian discord DG(A : B) as α → ∞,Q.E.D.

To be rigorous, we did not prove that the DNG(A : B)is lower bounded by DG(A : B), and we cannot rule outthe possibility that DNG(A : B) < DG(A : B) for intermediatevalues of α. However, our numerical data do not support thispossibility since we never observe DNG(A : B) < DG(A : B)and we expect that DNG(A : B) → DG(A : B) from above asα → ∞.

[1] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki,Rev. Mod. Phys. 81, 865 (2009).

[2] H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901(2001).

[3] L. Henderson and V. Vedral, J. Phys. A 34, 6899 (2001).[4] K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral,

arXiv:1112.6238, 2012 [Rev. Mod. Phys. (to be published)].[5] M. Gu, H. M. Chrzanowski, S. M. Assad, T. Symul, K. Modi,

T. C. Ralph, V. Vedral, and P. K. Lam, Nat. Phys. 8, 671(2012).

[6] V. Madhok and A. Datta, Int. J. Mod. Phys. B 27, 1345041(2013).

[7] A. Datta, A. Shaji, and C. M. Caves, Phys. Rev. Lett. 100,050502 (2008); A. Datta and A. Shaji, Int. J. Quantum Inform.9, 1787 (2011); A. Brodutch and D. R. Terno, Phys. Rev. A83, 010301 (2011); A. Al-Qasimi and D. F. V. James, ibid. 83,032101 (2011); G. Passante, O. Moussa, D. A. Trottier, andR. Laflamme, ibid. 84, 044302 (2011).

[8] B. Dakic et al., arXiv:1203.1629.[9] A. Ferraro and M. G. A. Paris, Phys. Rev. Lett. 108, 260403

(2012).

[10] R. Blandino, M. G. Genoni, J. Etesse, M. Barbieri, M. G. A.Paris, P. Grangier, and R. Tualle-Brouri, Phys. Rev. Lett. 109,180402 (2012).

[11] L. S. Madsen, A. Berni, M. Lassen, and U. L. Andersen, Phys.Rev. Lett. 109, 030402 (2012).

[12] B. Dakic, V. Vedral, and C. Brukner, Phys. Rev. Lett. 105,190502 (2010).

[13] S. Luo, Phys. Rev. A 77, 042303 (2008); M. Ali, A. R. P. Rau,and G. Alber, ibid. 81, 042105 (2010).

[14] P. Giorda and M. G. A. Paris, Phys. Rev. Lett. 105, 020503(2010).

[15] G. Adesso and A. Datta, Phys. Rev. Lett. 105, 030501 (2010).[16] G. Giedke and J. I. Cirac, Phys. Rev. A 66, 032316 (2002);

J. Fiurasek and L. Mista, Jr., ibid. 75, 060302(R) (2007).[17] R. Vasile, P. Giorda, S. Olivares, M. G. A. Paris, and

S. Maniscalco, Phys. Rev. A 82, 012313 (2010); G. L. Giorgi,F. Galve, and R. Zambrini, Int. J. Quantum Inform. 9, 1825(2011); L. A. Correa, A. A. Valido, and D. Alonso, Phys. Rev.A 86, 012110 (2012).

[18] G. L. Giorgi, F. Galve, G. Manzano, P. Colet, and R. Zambrini,Phys. Rev. A 85, 052101 (2012).

052328-11

PAOLO GIORDA, MICHELE ALLEGRA, AND MATTEO G. A. PARIS PHYSICAL REVIEW A 86, 052328 (2012)

[19] R. Tatham, L. Mista, Jr., G. Adesso, and N. Korolkova, Phys.Rev. A 85, 022326 (2012).

[20] M. Koashi and A. Winter, Phys. Rev. A 69, 022309 (2004);S. Olivares and M. G. A. Paris, Int. J. Mod. Phys. B 27, 1245024(2012).

[21] G. Adesso and D. Girolami, Int. J. Quantum Inform. 9, 1773(2011).

[22] A. Datta, Ph.D. thesis, The University of New Mexico Albu-querque, New Mexico (2008), arXiv:0807.4490.

[23] G. Giedke, M. M. Wolf, O. Kruger, R. F. Werner, andJ. I. Cirac, Phys. Rev. Lett. 91, 107901 (2003). Exten-sions of the formula to nonsymmetric states were proposedin J. Solomon Ivan, and R. Simon, arXiv:0808.1658, andP. Marian and T. A. Marian, Phys. Rev. Lett. 101, 220403 (2008).

[24] M. G. A. Paris, Phys. Lett. A 217, 78 (1996).[25] F. A. M. de Oliveira, M. S. Kim, P. L. Knight, and V. Buzek,

Phys. Rev. A 41, 2645 (1990).[26] H. P. Yuen, J. Opt. Soc. Am. B 3, P86 (1986).[27] M. S. Kim, F. A. M. de Oliveira, and P. L. Knight, Phys. Rev. A

40, 2494 (1989).[28] R. T. Hammond, Phys. Rev. A 41, 1718 (1990).[29] C. F. Lo, Phys. Rev. A 43, 404 (1991); M. M. Nieto, Phys. Lett.

A 229, 135 (1997).[30] More general non-Gaussian measurements are in principle

possible. For instance, one might consider also nonorthogonalnon-Gaussian POVMs. Notice that in the two-qubit case [43],it has been shown that orthogonal measurements are almostoptimal, i.e., the use of nonorthogonal measurements cannotlead to a significantly lower minimum.

[31] C. T. Lee, Phys. Rev. A 42, 4193 (1990).[32] A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs,

H. J. Kimble, and E. S. Polzik, Science 282, 706(1998); N. Lee, H. Benichi, Y. Takeno, S. Takeda,J. Webb, E. Huntington, A. Furusawa, ibid. 332, 330(2011).

[33] V. D’Auria, S. Fornaro, A. Porzio,S. Solimeno, S. Olivares, and M. G. A. Paris, Phys. Rev.Lett. 102, 020502 (2009).

[34] A. Agliati, M. Bondani, A. Andreoni, G. De Cillis, and M. G.A. Paris, J. Opt. B 7, 652 (2005).

[35] S. Olivares and M. G. A. Paris, Phys. Rev. Lett. 107, 170505(2011).

[36] M. Ohliger, V. Nesme, D. Gross, Y.-K. Liu, and J. Eisert,arXiv:1111.0853 (2011).

[37] L. Albano, D. F. Mundarain, and J. Stephany, J. Opt. B 4, 352(2002).

[38] A. Ferraro, S. Olivares, and M. G. A. Paris,arXiv:quant-ph/0503237.

[39] S. Luo and S. Fu, Phys. Rev. A 82, 034302 (2010).[40] D. Cavalcanti, L. Aolita, S. Boixo, K. Modi, M. Piani, and

A. Winter, Phys. Rev. A 83, 032324 (2011); S. Boixo, L. Aolita,D. Cavalcanti, K. Modi, and A. Winter, Int. J. Quantum Inform.9, 1643 (2011).

[41] T. Tufarelli, D. Girolami, R. Vasile, S. Bose, and G. Adesso,arXiv:1205.0251 (2012).

[42] M. Piani, Phys. Rev. A 86, 034101 (2012).[43] F. Galve, G. Giorgi, and R. Zambrini, Europhys. Lett. 96, 40005

(2011).

052328-12

Recommended