Total versus quantum correlations in a twomode Gaussian state

Jamal El Qars

1 Department of Physics,Faculty of Applied Sciences,Ait-Melloul,Ibn Zohr University,Agadir,Morocco

2 EPTHE,Department of Physics,Faculty of Sciences,Ibn Zohr University,Agadir,Morocco

3 LPHE-MS,Department of Physics,Faculty of Sciences,Mohammed V University,Rabat,Morocco

Abstract In Li and Luo (2007 Phys.Rev.A 76 032327),the inequalitywas identified as a fundamental postulate for a consistent theory of quantum versus classical correlations for arbitrary measures of totalT and quantumQ correlations in bipartite quantum states.Besides,Hayden et al(2006 Commun.Math.Phys.265 95) have conjectured that,in some conditions within systems endowed with infinite-dimensional Hilbert spaces,quantum correlations may dominate not only half of total correlations but total correlations itself.Here,in a two-mode Gaussian state,quantifyingT andQ respectively by the quantum mutual information IG and the entanglement of formation(EoF)we verify thatis always less thanwhen IG and are defnied via the Rényi-2 entropy.While via the von Neumann entropy,may even dominate itself,which partly consolidates the Hayden conjecture,and partly,provides strong evidence hinting that the origin of this counterintuitive behavior should intrinsically be related to the von Neumann entropy by which the EoFis defined,rather than related to the conceptual definition of the EoFEF.The obtained results show that-in the special case of mixed two-mode Gaussian statesquantum entanglement can be faithfully quantified by the Gaussian Rényi-2 EoF

Keywords: entanglement,von Neumann and Rényi-2 entropies,quantum mutual information,Gaussian states,spontaneous emission laser

1.Introduction

In a quantum bipartite statetotal correlationsT can be divided into two parts: classicalC and quantumQ,then T = Q +C [1].Besides,adopting the Henderson-Vedral conjecture [1],that is,classical correlations should not be less than quantum ones,i.e.C ≥ Q,it follows thatwhich was identified as a fundamental postulate for a consistent theory of quantum versus classical correlations in bipartite states for arbitrary measures of quantum Q and totalT correlations[2].

On the one hand,it is generally accepted that the quantum mutual information-denoted I-is the information-theoretic measure of total correlationsT in quantum bipartite states[3-5].Before being introduced in [6],the concept of quantum mutual information has been implicitly used some years ago to study information transfer in quantum measurements [7] and subsequently rediscovered in[8].From an operational point of view,the quantum mutual information was first interpreted as the amount of randomness noise needed to erase completely the correlations in a quantum stateby turning it into a product state,i.e.[3].Later,it was rigorously associated with the maximum amount of information that can be securely transmitted between a sender X and a receiverY,who share a correlated state[4].

On the other hand,quantum correlations of mixed states,can arise in diverse incarnations,i.e.Bell nonlocality [9],steering [10],entanglement [11],and discord [5,12],where all can be exploited for enhancing information processing tasks over any classical approach [13].In particular,entanglement-usually used as synonymous of quantum correlations-was systematically defined as a kind of nonseparable quantum correlations that cannot be prepared only by means of local operations and classical communication [14].Nowadays,it becomes evident that entanglement,as a fundamental physical resource for quantum protocols,can play a crucial role in quantum information science [15].

One of the central tasks of quantum information theory is to quantify the amount of entanglement that a quantum state possesses [16].In pure bipartite states,there is a universal bona fide quantifier of entanglement,i.e.the entropy of entanglement [17].However,in mixed states-where a more complex scenario emerges-the entropy of entanglement no longer deserves to be a measure of entanglement.Therefore,miscellaneous entanglement measures,which can be distinguished due to their operational meaning or mathematical structures,have been introduced [16].

From a conceptual point of view,Gaussian entanglement measures can exhaustively be classified into two categories.The first one encompasses the negativities[18-20],while,the second is provided by the so-called convex-roof extended measures [21].Except for the entanglement of formation(EoF) [22]-which physically interpreted as the minimal entanglement needed for preparing an entangled state by mixing pure entangled states,and mathematically computed for two-qubit states [23] as well as for arbitrary two-mode Gaussian states (TMGSs) [24,25]-no such measure is currently known [26].

Defining the quantum mutual information I and the entanglementE by means of the von Neumann entropy,it has been shown in [2],within discrete-variable systems,that the inequalityis well satisfied when substitutingE by either the distillable entanglementED[22]or the squashed entanglementEsq[27].While,it may be violated with respect to the EoFEFor the entanglement costEC[22],where the former may exceed not only half of total correlations,but total correlations itself [2].In a similar context,it has been shown for a two-qubit system,that quantum correlations quantified by means of the von Neumann EoFEFmay exceed classical ones [28],which violates the Henderson-Vedral conjecture[1].Consequently-as main conclusion pointed out in [2]-the EoFEFis too big to be regarded as a genuine measure of entanglement,in the sense that it may exceed total correlations.Attempting to explain the origin of the peculiar behaviors exhibited by the EoFEFin[2,28],some doubts around the validity of the Henderson-Vedral conjecture as well as the validity of the pure-state decompositions in the general definition of the EoFEFhave seriously been raised in [2,28].

Figure 1.A nondegenerate three-level laser in a cascade configuration [32].The transition |a〉→|b〉(|b〉→|c〉) of frequency ω1(ω2)and spontaneous emission decay rate γab(γbc) is assumed to be dipole-allowed,while,|a〉→|c〉 is dipole forbidden [29].

In the past decades,a correlated spontaneous emission laser-in which nondegenerate three-level atoms in a cascade configuration are injected into a cavity in a coherent superposition-has attracted a special attention in connection with its potential as a source of a highly correlated two-mode Gaussian light [29,30].The quantum correlations between the emitted photons can be induced by preparing the atoms,initially,in a coherent superposition of the upper and lower levels [31-33] or by coupling these two levels by a strong coherent external driving [34] or also by using the two processes simultaneously [35].Essentially,a correlated spontaneous emission laser is believed to be a source of strong quantum correlations [29,30],which therefore can be employed for testing quantum nonlocality[36-39].Here,this system is chosen as a viable and reliable scheme for investigating the Hayden conjecture [40] and the constraintin a TMGS.

Hence,in a mixed Gaussian stateinvolving two cavity modes of a nondegenerate three-level laser[29,41-43],and motivated by the fact that the inequalityis proven to hold for general mixed TMGSs [44,45],where I andEFare defined via the Rényi-2 entropy[46],we give strong evidence hinting that the origin of the counterintuitive behaviors exhibited by the EoFEFin[2,28],should intrinsically be related to the von Neumann entropy by which the EoFEF,Vis defined,rather than related to the conceptual definition of the EoFEFor the Henderson-Vedral conjecture.For this,we first verify the holding of the inequalitynext,via the von Neumann entropy,we show that evenmay happen,which largely violatesand further supports the Hayden conjecture [40].

The remainder of this paper is organized as follows.In section 2,we introduce the system at hand,and we derive the master equation for the stateNext,we obtain the dynamics of the first and second moments of the two cavity modes variables,and further we evaluate the stationary covariance matrix of the stateIn section 3,we quantify the EoF as well as the quantum mutual information in the bipartite stateusing two different entropic measures,i.e.the von Neumann and the Rényi-2 entropies.Also,we present our results.Finally,in section 4,we draw our conclusions.

2.Model and master equation

In a cavity coupled to a vacuum reservoir,we consider an ensemble of nondegenerate three-level atoms resonantly interacting with two cavity modes of the quantized cavity field.The jth cavity mode is specified by its annihilation operatorfrequency ωjand a decay rate κj(we take for simplicity κ1,2=κ).The atoms are assumed to be injected into the cavity at a rate raand removed within a time τ [31].As shown in figure 1,the upper,intermediate and lower energy levels of a single atom are denoted respectively by|a〉,|b〉 and |c〉.

In the rotating wave approximation,the interaction between the two cavity modes and a single three-level atom,can be described in the interaction picture by the Hamiltonian[30]

with χab(χbc) being the coupling constant for the transition|a〉→|b〉(|b〉→|c〉)[31].Also,we assume that the atoms are initially prepared in an arbitrary coherent superposition of the upper |a〉 and the lower |c〉 energy levels with the probabilities |α|2and |β|2[33].The initial state of a single atom writes |Ψatom(0)〉=α|a〉+β|c〉,thus,its associated density operator is given by

The master equation for the reduced density operatorof the two cavity modes,writes [47,48]

We need to carry out our study in the stationary regime,then,the non-zero steady-state solutions of equations ((5)-(9)) can be expressed as

Notice that the equations ((10)-(12)) are physically meaningful only if η ≥0,so that 0 ≤η ≤1,which corresponds to the regime of lasing without population inversion [30].

where the block-matrix σj=?jj1l2represents the jth cavity mode for j=1,2,while σ3=?12diag(1,-1) describes the correlations between them,withand

3.EoF versus total correlations in a two-mode Gaussian state

3.1.Gaussian EoF

Before defining the Gaussian EoF,we briefly recall what is intended by entanglement.A bipartite pure state∣ψXY〉is said to be entangled,if cannot be factorised as∣ψXY〉=∣φX〉 ?∣φY〉.While,a mixed stateis entangled if cannot be factorised as a convex combinations of product states,i.e.where piare probabilities with ∑ipi=1 [11].

In pure bipartite states,the entropy of entanglement is a simple and unique measure of entanglement[17].However,for mixed states,such a measure no longer deserves for quantifying entanglement[11],and therefore various measures,including the Rényi-α Gaussian EoF,were introduced [24,25,44].

For a bipartite quantum statethe Ré nyi-α EoFEF,αis defined as the convex-roof of the Rényi-α entropylnon pure states [44,50],i.e.

where the minimisation is over all the decompositions ofinto set of pure states {|ψi〉} with pi≥0 and ∑ipi=1.

Here,let us pause to briefly recall that-in quantum information theory-the degree of information possessed by a bipartite quantum stateis conventionally quantified by the von Neumann entropy defined asthat is the direct counterpart to the Shannon entropy in classical information theory [51].More precisely,such entropy-used under the name of entanglement entropy-quantifies the degree of quantum information contained in an ensemble of a large number of independent and identically distributed copies of the state [52].In addition,it was widely employed to study entanglement in miscellaneous fields of physics,e.g.ground states of quantum many body systems and lattice systems [53,54],relativistic quantum field theory[55],and the holographic theory of black holes [56].

Besides,the α-Rényi entropies defined as(1 -α)-1lnwith α ∈(0,1)∪(1,+∞) [46],were introduced as a generalization of the von Neumann entropy[56,57].Their interpretation is essentially related to derivatives of the free energy with respect to temperature [58],and which have found applications,particularly,in the study of channel capacities [59],work value of information [60],and entanglement spectra in many-body systems [61].

Notice that the class of α-Rényi entropies are continuous,non-negatives,invariants under the action of the unitary operations,additive on tensor-product states,and converge to the von Neumann entropy in the limit α →1 [62].

Given an arbitrary TMGS with covariance matrixσXY,an upper bound of the Rényi-α EoFEF,αcan be obtained by limiting the decomposition in equation(14)only over pure Gaussian states,therefore,we obtain the Rényi-α Gaussian EoF [44,50]

where the minimisation is taken over a pure TMGS with covariance matrix ΞXYsmaller thanσXY,and the sub-matrix ΞXis the marginal covariance matrix of the first mode obtained from ΞXYby partial tracing over the second mode.

ChoosingαS to be the conventional von Neumann entropy defined,in the limit α →1,as[50],then equation (15) defines the usual Gaussian EoF[23-25].Finding analytical solution of the optimization problem in equation(15)for generic states is-in general-a nontrivial task,nevertheless,for α →1,closed formulas were determined for two-qubit states[23],isotropic and Werner states [63,64],as well as for arbitrary TMGSs [24,25].

Within the Gaussian framework,it has been demonstrated that the Rényi-2 entropyis operationally linked to the Shannon entropy of Gaussian Wigner distributions sampling by homodyne detections in phase space[50],in addition,it fulfills the strong subadditivity inequality,i.e.[50],a key requirement for quantum information theory,therefore,it can be used for defining bona fide measures of correlations,such as the Rényi-2 Gaussian EoF [44,50] and the Rényi-2 Gaussian quantum mutual information [45,50].

In equation (15),replacingαS with the Rényi-2 entropy S2,we thus obtain the Rényi-2 Gaussian entanglement[50],which recently dubbed more fittingly as the Rényi-2 Gaussian EoF≡[44].

For two-mode squeezed thermal state with the covariance matrix σ12(13),the Gaussian EoFreads [24,25]

where s±=(?11±?22)/2 andg=

Notice that the asymptotic regularization of the EoFEF(i.e.coincides with the entanglement cost[22,66],an entanglement measure which was interpreted as the minimum amount of singlets (i.e.maximally entangled antisymmetric two-qubit states) needed for generating an entangled bipartite stateby means of local operations and classical communication [65,66].In fact,obtaining analytical expression of the entanglement costECfor an arbitrary state,is a difficult task [66].However,since the additivity of the EoFfor TMGSs is proven to be true[25],it consequently follows that[67].

Finally,we recall some of interesting properties of the Rényi-2 Gaussian EoF[45,50]: (i) it does not increase under all Gaussian local operations and classical communication,thus it is a proper measure of Gaussian entanglement; (ii) it is additive on tensor product states; and (iii) it satisfies both the Coffman-Kundu-Wootters-type monogamy inequality [68] and the Koashi-Winter monogamy relation [69].

3.2.Gaussian quantum mutual information

Based on the Landauer’s erasure principle [70],Groisman et al [3] have defined the quantum mutual information as the amount of noise required to erase completely the correlations contained in a joint density operatorby turning it into a product state,i.e.Essentially,a strong argument in favor for accepting the quantum mutual information as a measure of total correlations in bipartite states is given in [4],i.e.if two parties X andY share a quantum correlated statethe maximum amount of information that X(Y) can securely send toY(X) is exactly equal to the quantum mutual information of the shared state

From an operational point of view,the von Neumann quantum mutual information given by equation (18) quantifies the amount of the information extracted onby looking at the system in its entirety,minus the information that can be obtained from separate observations of the subsystems with the marginal statesandIt is always positive and vanishes only if[3-5].

where f(x) is defined above,Ik= detσkandare respectively the symplectic invariants and the symplectic eigenvalues of the covariance matrix σ12(13) with Δ=I1+I2+2I3.Whereas,by means of the Gaussian Rényi-2 entropythe Rényi-2 quantum mutual informationreads [50]

which shown to coincide exactly with the Shannon continuous mutual information of the Wigner function of the TMGS[50].Operationally,the Rényi-2 Gaussian quantum mutual informationcan be interpreted as the amount of extra discrete information that needs to be transmitted through a continuous variable channel for reconstructing the complete joint Wigner function of the TMGSrather than just the two marginal Wigner functions of the subsystems [50,71].In other words,quantifies the total quadrature correlations of the state[50].Finally,we notice that the Rényi-2 quantum mutual informationis always non-negative for Gaussian states,and vanishes only ifhowever,it can be negative for more general states (e.g.non-Gaussian ones),then,it does not admit an operational interpretation beyond the Gaussian framework.

In what follows,we show that-under various circumstances-the constraintis well satisfied via the Rényi-2 entropy,while,it can largely be violated via the von Neumann entropy.

In figure 2,we plot(panel (a)) and(panel (b)) against the population inversion η for various values of the linear gain coefficient A,with a cavity decay rate κ=1.Remarkably,figure 2(a) shows that for a density values of η and A,is always positive meaning that the inequalityis well satisfied when the quantum mutual informationand the EoFare defined via the Rényi-2 entropy.Whereas,in spite of the fact that the entanglement degree is generally accounts as a portion of total correlations,figure 2(b) shows thatmay be negative,thenmay exceedwhich constitutes a large violation of the constraintSuch counterintuitive behavior (i.e.supporting the Hayden conjecture [40],clearly evidences that the von Neumann EoFmay not be the best choice for estimating entanglement in mixed TMGSs,which is quite consistent with the conclusion pointed out in [20].

Next,in figure 3,we plot(panel(a))and(panel (b)) against the parameters κ and A for η=0.2.Similarly to the results illustrated in figure 2,figure 3(a) shows within a density values of κ and A that the constraintis well satisfied with respect to the Rényi-2 entropy.While,another violation ofvia the von Neumann entropy is shown in figure 3(b),where evenmay happen,which undermines the interpretation of the von Neumann EoFEF,Vas just a fraction of the total correlations.

Figure 2.(a):and (b):versus the parameters A and η for κ=1.(a) shows that the constraint (1 2) is satisfied via the Rényi-2 entropy(i.e.α=2),in contrast,it is largely violated-in(b)-via the von Neumann entropy(i.e.α=1),where even may happen,which consolidates the Hayden conjecture[40],in addition,undermines the interpretation of the von Neumann EoF EF ,Vas just a fraction of the total correlations.

Here,it is interesting to emphasize that the holding of the constraintfor a wide range of the parameters κ,A and η in figures 2(a)-3(a),is not a curious coincidence since,the inequalityis proven to be true as long as α ≥2 for general mixed TMGSs[44,45].Howeveras can be seen from figures 2(b)-3(b)-it may be largely violated with respect to the von Neumann entropy (α=1),where evenmay happen,which therefore provides strong evidence hinting that the origin of this counterintuitive behavior should be related to the von Neumann entropy by which the EoFEF,Vis defined,rather than related to the conceptual definition of the EoFEFor the Henderson-Vedral conjecture.

In quantum information theory,the additivity is a very desirable property that can largely reduce the evaluation of entanglement [20].Indeed,since quantum mechanics is statistical,often physical meaning of entanglement measures is acquired only in the asymptotic regime of many copies of a given state,which can be reduced to a single copy for additive measures [72],such as the conditional entanglement of mutual information [72],the logarithmic negativity [19],the squashed entanglement [27],the EoFEF,Vof arbitrary TMGSs [25],and the entanglement cost [66].

Therefore,since the additivity of the von Neumann EoFis proven to be true,i.e.where ρ andρ′ are two entangled TMGSs [25],it follows that the entanglement costcoincides with the EoF[66,67],and consequently,both of them cannot be regarded as genuine measures of entanglement,in the sense that they may exceed total correlations[40].By contrast,th e Rényi-2 EoFseems more appropriate for estimating entanglement in mixed TMGSs,in the sense that,it satisfies the constraintin addition,it is additive and monogamous,which are two strong points for considering it as an entanglement measure [45,50].

Finally,we notice that the constrainthas also been violated in systems endowed with finite-dimensional Hilbert spaces [2,28],where the quantum mutual information I and the EoFEFare defined only via the conventional von Neumann entropy.Attempting to explain the origin of such a violation,some doubts around the validity of the pure-state decompositions in the general definition of the EoFEFand the validity of the Henderson-Vedral conjecture have seriously been raised in [2,28].However,the comparative study accomplished here,provides strong evidence suggesting that the origin of the counterintuitive behavior exhibited by the EoFEFin figures 2(b)-3(b) as well as in[2,28] should intrinsically be related to the von Neumann entropy by which the EoFEF,Vis defined,rather than related to the conceptual definition of the EoFEFor the Henderson-Vedral conjecture.More generally,the results obtained here evidence that the Gaussian EoFdefined via the von Neumann entropy may not be the best choice for estimating entanglement in mixed TMGSs,which is quite consistent with [20].

4.Conclusion

In a TMGS,we studied the inequalitythat was considered as a fundamental postulate for a consistent theory of quantum versus classical correlations for arbitrary measures of totalT and quantumQ correlations [2].Using the Gaussian EoFand the Gaussian quantum mutual informationrespectively as measures of quantumQ and total correlationsT for α=1,2,we verified thatis well satisfied whenandare defined via the Rényi-2 entropy (i.e.α=2).In contrast,via the von Neumann entropy (i.e.α=1),evenmay happen,which consolidates the Hayden conjecture [40],in addition,clearly evidences that the von Neumann EoFEF,Vmay not be the best choice for estimating entanglement in mixed TMGSs.Moreover,since the additivity of the Gaussian EoFis proven to be true [25],which implies that the entanglement costand the EoFare identical[67],it follows that bothandcan not be regarded as genuine measures of entanglement in mixed TMGSs,in the sense that,they may dominate total correlations.

The comparative study accomplished here,provides strong evidence hinting that the origin of the peculiar behavior exhibited by the EoFEF,Vin figures 2(b)-3(b)as well as in[2,28],should intrinsically be related to the von Neumann entropy by which the EoFEF,Vis defined,rather than related to the conceptual definition of the EoFEFor the Henderson-Vedral conjecture[2,28].Moreover,it shows that the Rényi-2 EoFis more faithful-than the von Neumann EoF-for quantifying entanglement in mixed TMGSs,in the sense that,it satisfies the constraintfurthermore,it is additive and monogamous,which are two strong points for considering it as an entanglement measure [45,50].

Finally,it is interesting to mention that apart from total correlation-that can be written as sum of quantum and classical parts in bipartite states-there exist similar equalities in the contexts of coherence and randomness,where the total amount of a property can also be decomposed into classical and quantum parts[73,74].In this respect,we believe that the full understanding of the relationships between quantum and classical quantities contained in bipartite mixed states is of critical importance,which represents a relevant step towards efficient characterization and quantification of quantum entanglement.This therefore would open up the application of the theoretical work on quantum information processing and communication.

Acknowledgments

I am particularly indebted to an anonymous referee for constructive critiques and insightful comments.