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Extremal entanglement and mixedness in continuous variable systems

Gerardo Adesso, Alessio Serafini, and Fabrizio IlluminatiDipartimento di Fisica “E. R. Caianiello”, Universita di Salerno, INFM UdR di Salerno,

INFN Sezione di Napoli, Gruppo Collegato di Salerno, Via S. Allende, 84081 Baronissi (SA), Italy(Dated: May 7, 2004)

We investigate the relationship between mixedness and entanglement for Gaussian states of continuous vari-able systems. We introduce generalized entropies based on Schattenp-norms to quantify the mixedness of astate, and derive their explicit expressions in terms of symplectic spectra. We compare the hierarchies of mixed-ness provided by such measures with the one provided by the purity (defined astr 2 for the state ) for genericn-mode states. We then review the analysis proving the existence of both maximally and minimally entangledstates at given global and marginal purities, with the entanglement quantified by the logarithmic negativity.Based on these results, we extend such an analysis to generalized entropies, introducing and fully characterizingmaximally and minimally entangled states for given global and local generalized entropies. We compare thedifferent roles played by the purity and by the generalizedp-entropies in quantifying the entanglement and themixedness of continuous variable systems. We introduce theconcept of average logarithmic negativity, showingthat it allows a reliable quantitative estimate of continuous variable entanglement by direct measurements ofglobal and marginal generalizedp-entropies.

PACS numbers: 03.67.-a, 03.67.Mn, 03.65.Ud

I. INTRODUCTION

The degree of entanglement (i.e. the contents inquantumcorrelations) as well as the degree of mixedness (i.e. “theamount” by which a state fails to be pure) are among the cru-cial features of quantum states from the point of view of quan-tum information theory. Indeed, the search for proper analyt-ical ways to quantify such features for general (mixed) quan-tum states cannot be yet considered accomplished. In view ofsuch considerations, it is clear that the full understanding ofthe relationships between the quantum correlations containedin a bipartite state and the global and local (i.e.referring to thereduced states of the two subsystems) degrees of mixedness ofthe state, would be desirable. In particular, it would be a rel-evant step towards the clarification of the nature of quantumcorrelations and, possibly, of the distinction between quan-tum and classical correlations of mixed states, which remainsan open issue [1]. A simple question one can raise in sucha context is the investigation of the properties of extremallyentangled states for a given degree of mixedness.

Let us mention that, as for two–qubit systems, the notionof maximally entangled states at fixed mixedness (MEMS)was originally introduced by Ishizaka and Hiroshima [2]. Thediscovery of such states spurred several theoretical works[3], aimed at exploring the relations between different mea-sures of entanglement and mixedness [4] (strictly related tothe question of theorderingof these different measures [5]).Moreover, maximally entangled states for given local (or“marginal”) mixednesses have been recently introduced andanalyzed in detail in the context of qubit systems [6]. Onthe experimental side, much attention has been devoted toexploring the two-qubit Hilbert space in optical settings [7],while the experimental realization of MEMS has been re-cently demonstrated [8].

Because of the great current interest in continuous variable(CV) quantum information [9, 10, 11, 12], the extension ofsuch analyses to infinite dimensional systems is higly desir-

able. In the present work, we introduce and study in detailextremally entangled mixed Gaussian states of infinite dimen-sional Hilbert spaces for fixed global and marginal generalizedentropies, significantly generalizing the results derivedearlierin Ref. [13], where the existence of maximally and minimallyentangled mixed Gaussian states at given global and marginalpurities was first discovered. In the present paper we willmake use of a hierarchy of generalized entropies, based onthe Schattenp-norms, to quantify mixedness and character-ize extremal entanglement in continuous varibale (CV) sys-tems and investigate several related subjects, like the orderingof such different entropic measures and the relations betweenEPR (Einstein-Podolsky-Rosen) correlations and symplecticspectra. The crucial starting point of our analysis is the obser-vation that the existence of infinitely entangled states [14, 15]prevents maximally entangled Gaussian states from being de-fined as states with maximal, finite, logarithmic negativity,even at fixed global mixedness [16]. However, we will showthat fixing the globaland the local mixednesses allows to de-fine unambiguously both maximally and minimally entangledmixed Gaussian states. This relevant and somehow surprisingresult – there is no analog of Gaussian minimally entangledstates in finite dimensional systems – turns out to have an ex-perimental interest as well [13, 17].

The paper is structured as follows. In Sec. II we briefly re-view the basic notation and the general properties of Gaussianstates. In Sec. III we introduce the hierarchy of generalizedSchattenp-norms and entropies, and we extensively discussthe problem of the ordering of different entropic measuresfor states with an arbitrary number of modes. In Sec. IV wereview the state of the art on the existing, computable mea-sures of entanglement for two–mode Gaussian states, whilein Sec. V we present a heuristic argument relating EPR cor-relations to symplectic spectra. In Sec. VI we introduce aparametrization of two–mode Gaussian states in terms of sym-plectic invariants endowed with a direct physical interpreta-tion. Exploiting these results, in Sec. VII we define maximallyand minimally entangled states for given global and marginal

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purities and present some experimental situations in whichthese states occur. In Sec. VIII we generalize the concept ofextremal entanglement in CV systems by introducing max-imally and minimally entangled states for given global andlocal generalized entropies of arbitrary order, and we presentan extensive study of their properties. We find out, somehowsurprisingly, that maximally and minimally entangled statescan interchange their roles for certain ranges of values of theglobal and marginalp-entropies; moreover, we observe thatwith increasingp the generalized entropies, while carrying ingeneral less information on a quantum state, provide a moreaccurate quantification of its entanglement. In Sec. IX we in-troduce and study the concept of “average logarithmic nega-tivity”, showing that this quantity provides an excellent quan-titative estimate of CV entanglement based only on the knowl-edge of global and local entropies. Finally, in Sec. X we sum-marize our results and discuss future perspectives.

II. GAUSSIAN STATES: GENERAL OVERVIEW

Let us consider a CV system, described by an Hilbert spaceH =

⊗ni=1 Hi resulting from the tensor product of the infi-

nite dimensional Fock spacesHi’s. Let ai be the annihilationoperator acting onHi, andxi = (ai+a†i) andpi = (ai−a†i )/ibe the related quadrature phase operators. The correspondingphase space variables will be denoted byxi andpi. Let usgroup together the operatorsxi and pi in a vector of opera-tors X = (x1, p1, . . . , xn, pn). The canonical commutationrelations for theXi’s are encoded in the symplectic formΩ

[Xi, Xj ] = 2iΩij ,

with Ω ≡n⊕

i=1

ω , ω ≡(

0 1−1 0

)

.

The set of Gaussian states is, by definition, the set of stateswith Gaussian characteristic functions and quasi–probabilitydistributions. Therefore a Gaussian state is completelycharacterized by its first and second statistical moments,which form, respectively, the vector of first momentsX ≡(

〈X1〉, 〈X1〉, . . . , 〈Xn〉, 〈Xn〉)

and the covariance matrix of

elementsσ

σij ≡1

2〈XiXj + XjXi〉 − 〈Xi〉〈Xj〉 , (1)

where, for any observableo, the expectation value〈o〉 ≡Tr(o). First statistical moments can be arbitrarily adjustedby local unitary operations, which cannot affect any propertyrelated to entanglement or mixedness. Therefore they will beunimportant to our aims and we will set them to0 in the fol-lowing, without any loss of generality. Throughout the paper,σ will stand for the covariance matrix of the Gaussian state.

Let us now consider the hermitian operatory = Y XT ,whereY ∈ R

2n is an arbitrary real2n-dimensional row vec-tor. Positivity of imposesTr(y2) ≥ 0, which can be sim-ply recast in terms of second moments asY τY T ≥ 0, with

τij = 〈XiXj〉. From this relation, exploiting the canonicalcommutation relations and recalling definition (1) and the ar-bitrarity of Y , the Heisenberg uncertainty principle can be re-cast in the form

σ + iΩ ≥ 0 , (2)

Inequality (2) is the necessary and sufficient constraintσ hasto fulfill to be a bona fidecovariance matrix [18, 19]. Wemention that such a constraint impliesσ ≥ 0.

In the following, we will make use of the Wigner quasi–probability representationW (xi, pi) defined, for any densitymatrix, as the Fourier transform of the symmetrically orderedcharacteristic function [20]. In Wigner phase space picture,the tensor productH =

⊗Hi of the Hilbert spacesHi’s ofthen modes results in the direct sumΓ =

⊕

Γi of the phasespacesΓi’s. In general, as a consequence of the Stone-vonNeumann theorem, any symplectic transformation acting onthe phase spaceΓ corresponds to a unitary operator acting onthe Hilbert spaceH, through the so calledmetaplecticrep-resentation. Such a unitary operator is generated by termsof the second order in the field operators [19]. In what fol-lows we will refer to a transformationSl =

⊕

Si, with eachSi ∈ Sp(2,R) acting onΓi, as to a “local symplectic opera-tion”. The corresponding unitary transformation is the “localunitary transformation”Ul =

⊗

Ui, with eachUi acting onHi.

The Wigner function of a Gaussian state can be written asfollows in terms of phase space quadrature variables

W (X) =e−

12Xσ

−1XT

π√Detσ

, (3)

whereX stands for the vector(x1, p1, . . . , xn, pn) ∈ Γ.Finally let us recall that, due to Williamson theorem [21],

the covariance matrix of an–mode Gaussian state can alwaysbe written as [18]

σ = STνS , (4)

whereS ∈ Sp(2n,R) andν is the covariance matrix

ν = diag(ν1, ν1, . . . , νn, νn) , (5)

corresponding to a tensor product of thermal states with diag-onal density matrix ⊗ given by

⊗ =⊗

i

2

νi + 1

∞∑

k=0

(

νi − 1

νi + 1

)

|k〉ii〈k| , (6)

|k〉i being thek-th number state of the Fock spaceHi. Thedual (Hilbert space) formulation of Eq. (4) then reads: =U † ⊗ U , for some unitaryU .The quantitiesνi’s form the symplectic spectrum of the co-variance matrixσ and can be computed as the eigenvalues ofthe matrix|iΩσ|. Such eigenvalues are in fact invariant underthe action of symplectic transformations on the matrixσ.The symplectic eigenvaluesνi encode essential informationson the Gaussian stateσ and provide powerful, simple ways to

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express its fundamental properties. For instance, let us con-sider the Heisenberg uncertainty relation (2). SinceS is sym-plectic, one hasS−1T

ΩS−1 = Ω, so that inequality (2) isequivalentto ν + iΩ ≥ 0. In terms of the symplectic eigen-valuesνi the uncertainty relation then simply reads

νi ≥ 1 . (7)

We can, without loss of generality, rearrange the modes of an-mode state such that the corresponding symplectic eigen-values are sorted in ascending order

ν1 ≤ ν2 ≤ . . . ≤ νn−1 ≤ νn .

With this notation, the uncertainty relation reduces toν1 ≥ 1.We remark that the full saturation of Heisenberg uncertaintyprinciple can only be achieved by puren-mode Gaussianstates, for whichνi = 1 ∀i = 1, . . . , n. Instead, mixedstates such thatνi≤k = 1 andνi>k > 1, with 1 ≤ k ≤ n,only partially saturate the uncertainty principle, with partialsaturation becoming weaker with decreasingk. Such statesare minimum uncertainty mixed Gaussian states in the sensethat the phase quadrature operators of the firstk modes satisfythe Heisenberg minimal uncertainty, while for the remainingn − k modes the state indeed contains some additional ther-mal and/or Schrodinger–like correlations which are responsi-ble for the global mixedness of the state.

A. Two–mode states

In the present work we will mainly deal with two–modeGaussian states. Here, we will thus briefly review their rele-vant properties and specify some further notations.

The expression of the two–mode covariance matrixσ interms of the three2× 2 matricesα, β, γ will be useful

σ ≡(

α γ

γT β

)

. (8)

It is well known that for any two–mode covariance matrixσ

there exists a local symplectic operationSl = S1 ⊕ S2 whichtakesσ to the so called standard formσsf [22, 23]

STl σSl = σsf ≡

a 0 c+ 00 a 0 c−c+ 0 b 00 c− 0 b

. (9)

States whose standard form fulfillsa = b are said to be sym-metric. Let us recall that any pure state is symmetric andfulfills c+ = −c− =

√a2 − 1. The correlationsa, b, c+,

andc− are determined by the four local symplectic invariantsDetσ = (ab − c2+)(ab − c2−), Detα = a2, Detβ = b2,Detγ = c+c−. Therefore, the standard form correspondingto any covariance matrix is unique.

The uncertainty principle Ineq. (2) can be recast as a con-straint on theSp(4,R) invariantsDetσ and∆(σ) = Detα +Detβ + 2Detγ:

∆(σ) ≤ 1 + Detσ . (10)

Let us mention that, as it is evident from Eq. (9), the conditionσ ≥ 0 implies

ab− c2∓ ≥ 0 . (11)

The symplectic eigenvalues of a two–mode Gaussian statewill be namedν− andν+, with ν− ≤ ν+ in general. Withsuch an ordering, the Heisenberg uncertainty relation Eq. (7)becomes

ν∓ ≥ 1 . (12)

Full saturationν− = ν+ = 1 yields the standard pure Gaus-sian states of Heisenberg minimum uncertainty; partial satura-tion ν− = 1, ν+ > 1 defines the minimum-uncertainty mixedGaussian states. A simple expression for theν∓ can be foundin terms of the twoSp(4,R) invariants [24, 25]

2ν2∓ = ∆(σ)∓√

∆(σ)2 − 4Detσ . (13)

In turn, Eq. (13) yields immediately

∆(σ) = ν2− + ν2+ . (14)

A subclass of Gaussian states that will play a relevant rolein the following is constituted by the nonsymmetric two–modesqueezed thermal states. LetSr = exp(12ra1a2 − 1

2ra†1a

†2)

be the two mode squeezing operator with real squeezing pa-rameterr, and let⊗

νi be a tensor product of thermal stateswith covariance matrixνν∓ = 12ν− ⊕ 12ν+, whereν∓ is,as usual, the symplectic spectrum of the state. Then, a non-symmetric two-mode squeezed thermal stateξνi,r is definedasξνi,r = Sr

⊗νiS

†r , corresponding to a standard form with

a = ν− cosh2 r + ν+ sinh2 r ,

b = ν− sinh2 r + ν+ cosh2 r , (15)

c± = ±ν− + ν+2

sinh 2r .

We mention that the peculiar entanglement properties ofsqueezed nonsymmetric thermal states have been recently an-alyzed [26]. In the symmetric instance (withν− = ν+ = ν)these states reduce to two–mode squeezed thermal states. Thecovariance matrices of these states are symmetric standardforms with

a = ν cosh 2r , c± = ±ν sinh 2r . (16)

In the pure case, for whichν = 1, one recovers the two–mode squeezed vacua. Notice that such states encompass allthe standard forms associated to pure states: any two–modeGaussian state can reduced to a squeezed vacuum by meansof unitary local operations.Two–mode squeezed states (both thermal and pure) are en-dowed with remarkable properties related to entanglement[27, 28, 29]. Even the ideal, perfectly correlated originalEPRstate [30] can be seen indeed as a two–mode squeezed vacuumin the limit of infinite squeezing parameterr. The dynamicalproperties of the entanglement and the characterization ofthedecoherence of such states have been addressed in detail inseveral works [31]. We will also show, as a byproduct of thepresent work, the peculiar role they play as maximally entan-gled Gaussian states.

4

III. MEASURES OF MIXEDNESS, DEGREE OFCOHERENCE, AND ENTROPIC MEASURES

The degree of mixedness of a quantum state can be char-acterized completely by the knowledge of all the associatedSchattenp–norms [32]

‖‖p ≡ (Tr ||p) 1p = (Tr p)

1p , with p ≥ 1. (17)

In particular, the casep = 2 is directly related to the purityµ = Tr 2 = (‖‖2)2 [33]. Thep-norms are multiplicativeon tensor product states and thus determine a family of “gen-eralized entropies”Sp [34, 35], defined as

Sp =1− Tr p

p− 1, p > 1. (18)

These quantities have been introduced independently by M.J. Bastiaans in the context of quantum optics [34], and byC. Tsallis in the context of statistical mechanics [35]. In thequantum arena, they can be interpreted both as quantifiers ofthe degree of mixedness of a state by the amount of in-formation it lacks, and as measures of the overall degree ofcoherence of the state (the latter meaning was elucidated byBastiaans in his analysis of the properties of partially coher-ent light). The quantityS2 = 1 − µ ≡ SL, conjugate tothe purityµ, is usually referred to as the linear entropy: itis a particularly important measure of mixedness, essentiallybecause of the simplicity of its analytical expressions, whichwill become soon manifest. Finally, another important classof entropic measures includes the Renyi entropies [36]

SRp =

ln Tr p

1− p, p > 1. (19)

It can be easily shown that

limp→1+

Sp = limp→1+

SRp = −Tr ( ln ) ≡ SV , (20)

so that also the Shannon-von Neumann entropySV can be de-fined in terms ofp-norms. The quantitySV is additive on ten-sor product states and provides a further convenient measureof mixedness of the quantum state.

It is easily seen that the generalized entropiesSp’s rangefrom 0 for pure states to1/(p − 1) for completely mixedstates with fully degenerate eigenspectra. Notice thatSV isinfinite on infinitely mixed states, whileSL is normalized to1. We also mention that, in the asymptotic limit of arbitrarylargep, the functionTr p becomes a function only of thelargest eigenvalue of: more and more information about thestate is discarded in such an estimate for the degree of purity;consideringSp in the limit p → ∞ yields a trivial constantnull function, with no information at all about the state underexam. We also note that, for any given quantum state,Sp is amonotonically decreasing function ofp.

Because of their unitarily invariant nature, the generalizedpuritiesTr p of genericn–mode Gaussian states can be sim-ply computed in terms of the symplectic eigenvaluesνi of σ.In fact, a symplectic transformation acting onσ is embodied

by a unitary (trace preserving) operator acting on, so thatTr p can be easily computed on the diagonal stateν. Oneobtains

Tr p =

n∏

i=1

gp(νi) , (21)

where

gp(x) =2p

(x+ 1)p − (x− 1)p.

A first consequence of Eq. (21) is that

µ() =1∏

νi=

1√Detσ

. (22)

Regardless of the number of modes, the purity of a Gaussianstate is fully determined by the symplectic invariantDetσalone. A second consequence of Eq. (21) is that, together withEqs. (18) and (20), it allows for the computation of the vonNeumann entropySV of a Gaussian state, yielding

SV () =

n∑

i=1

f(νi) , (23)

where

f(x) ≡ x+ 1

2ln

(

x+ 1

2

)

− x− 1

2ln

(

x− 1

2

)

.

Such an expression for the von Neumann entropy of a Gaus-sian state was first explicitly given in Ref. [37]. Let us remarkthat, clearly, the symplectic spectrum of single mode Gaus-sian states, which consists of only one eigenvalueν1, is fullydetermined by the invariantDetσ = ν21 Therefore, all the en-tropiesSp’s (andSV as well) are just increasing functions ofDetσ (i.e.of SL) and inducethe samehierarchy of mixednesson the set of one–mode Gaussian states. This is no longer truefor multi–mode states, even for the relevant, simple instanceof two–mode states.

Here we aim to find extremal values ofSp (for p 6= 2) forfixed SL in the generaln–mode Gaussian instance, in orderto quantitatively compare the characterization of mixednessgiven by the different entropic measures. For simplicity, incalculations we will replaceSL with µ. In view of Eqs. (21)and (22), the possible values taken bySp for a givenµ aredetermined by

(p− 1)Sp = 1−(

n−1∏

i=1

gp(si)

)

gp

(

1

µ∏n−1

i=1 si

)

, (24)

with 1 ≤ si ≤1

µ∏

i6=j sj. (25)

The last constraint on then− 1 real auxiliary parameterssi isa consequence of the uncertainty relation (7). We first focuson the instancep < 2, in which the functionSp is concavewith respect to anysi, for any value of thesi’s. Thereforeits minimum with respect to, say,sn−1 occurs at the bound-aries of the domain, forsn−1 saturating inequality (25). Since

5

Sp takes the same value at the two extrema and exploitinggp(1) = 1, one has

(p−1) minsn−1

Sp = 1−(

n−2∏

i=1

gp(si)

)

gp

(

1

µ∏n−2

i=1 si

)

. (26)

Iterating this procedure for all thesi’s leads eventually to theminimum valueSpmin(µ) of Sp at given purityµ, which sim-ply reads

Spmin(µ) =1− gp

(

1µ

)

p− 1, p < 2 . (27)

For p < 2, the mixedness of the states with minimal general-ized entropies at given purity is therefore concentrated inonequadrature: the symplectic spectrum of such states is partiallydegenerate, withν1 = . . . = νn−1 = 1 andνn = 1/µ.

The maximum valueSpmax(µ) is achieved by states satis-fying the coupled trascendental equations

gp

(

1

µ∏

si

)

g′p(sj) =1

µsj∏

sigp(sj)g

′p

(

1

µ∏

si

)

,

(28)where all the products

∏

run over the indexi from 1 ton− 1,and

g′p(x) =−p 2p

[

(x+ 1)p−1 − (x− 1)p−1]

[(x+ 1)p − (x− 1)p]2 . (29)

It is promptly verified that the above two conditions are ful-filled by states with a completely degenerate symplectic spec-trum: ν1 = . . . = νn = µ−1/n, yielding

Spmax(µ) =1− gp

(

µ− 1n

)n

p− 1, p < 2 . (30)

The analysis that we carried out forp < 2 can be straight-forwardly extended to the limitp → 1, yielding the extremalvalues of the von Neumann entropy for given purityµ of n–mode Gaussian states. Also in this case the states with max-imal SV are those with a completely degenerate symplecticspectrum, while the states with minimalSV are those with allthe mixedness concentrated in one quadrature. The extremalvaluesSVmin(µ) andSVmax(µ) read

SVmin(µ) = f

(

1

µ

)

, (31)

SVmax(µ) = nf(

µ− 1n

)

. (32)

The behaviors of the von Neumann and of the linear entropiesfor two–mode states are compared in Fig. 1.

The instancep > 2 can be treated in the same way, withthe major difference that the functionSp of Eq. (24) is convexwith respect to anysi for any value of thesi’s. As a conse-quence we have an inversion of the previous expressions: forp > 2, the states with minimalSpmin(µ) at given purityµ arethose with a fully distributed symplectic spectrum, with

Spmin(µ) =1− gp

(

µ− 1n

)n

p− 1, p > 2 . (33)

FIG. 1: Plot of the curves (solid lines) of maximal and minimal vonNeumann entropy at given linear entropy for two–mode Gaussianstates. The density of states in the plane of entropies is representedas well, by plotting the distribution of 20000 randomly generatedstates (dots).

FIG. 2: Plot of the absolute difference between the maximal and theminimal values of the generalized entropiesSp at fixed linear en-tropySL for two–mode Gaussian states, as a function ofp. Differentcurves correspond to different values of the linear entropy: SL = 0.8(dashed line),SL = 0.5 (continuous line), andSL = 0.1 (dash-dotted line).

On the other hand, the states with maximalSpmax at givenpurity µ are those with a spectrum of the kindν1 = . . . =νn−1 = 1 andνn = 1/µ. Therefore

Spmax(µ) =1− gp

(

1µ

)

p− 1, p > 2 . (34)

As shown in Fig. 2, the distance|Spmax−Spmin| decreaseswith increasingp. This is due to the fact that the quantitySp

carries less information with increasingp, and the knowledgeof µ provides a more precise bound on the value ofSp.

IV. MEASURES OF ENTANGLEMENT

We now review the main aspects of entanglement for CVsystems and discuss some of the possible quantifications of

6

quantum correlations for Gaussian states.The necessary and sufficient separability criterion for a

two–mode Gaussian statehas been shown to be positivityof the partially transposed state˜ (PPT criterion) [22]. In gen-eral, the partial transpositionof a bipartite quantum stateis defined as the result of the transposition performed on onlyone of the two subsystems in some given basis. Even thoughthe resulting˜ does depend on the choice of the transposedsubsystem and on the transposition basis, the statement˜≥ 0is geometric, and is invariant under such choices [38]. It canbe easily seen from the definition ofW (X) that the actionof partial transposition amounts, in phase space, to a mirrorreflection of one of the four canonical variables. In terms ofSp(2,R)⊕Sp(2,R) invariants, this reduces to a sign flip inDetγ

Det γ→ ˜−−−−→ −Detγ .

Therefore the invariant∆(σ) is changed into∆(σ) =∆(σ) = Detα + Detβ − 2Detγ. Now, the symplecticeigenvaluesν∓ of σ read

ν∓ =

√

√

√

√∆(σ)∓√

∆(σ)2 − 4Detσ

2. (35)

The PPT criterion thus reduces to a simple inequality thatmust be satisfied by the smallest symplectic eigenvalueν− ofthe partially transposed state

ν− ≥ 1 , (36)

which is equivalent to

∆(σ) ≤ Detσ + 1 . (37)

The above inequalities implyDet γ = c+c− < 0 as a nec-essary condition for a two–mode Gaussian state to be entan-gled. The quantityν− encodes all the qualitative characteri-zation of the entanglement for arbitrary (pure or mixed) two–modes Gaussian states. Note thatν− takes a particularly sim-ple form for entangled symmetric states, whose standard formhasa = b

ν− =√

(a− |c+|)(a− |c−|) . (38)

Inserting Eqs. (15) into Eq. (37) yields the following condi-tion for a two-mode squeezed thermal stateξνi,r to be entan-gled

sinh2(2r) >(ν2+ − 1)(ν2− − 1)

(ν− + ν+)2. (39)

As for the quantification of entanglement, no fully satisfac-tory measure is known at present for arbitrary mixed bipartiteGaussian states. However, a quantification of entanglementwhich can be computed for general Gaussian states is pro-vided by the negativityN , first introduced in Ref. [39], laterthoroughly discussed and extended in Ref. [24] to CV sys-tems. The negativity of a quantum state is defined as

N () =‖ ˜‖1 − 1

2, (40)

where˜ is the partially transposed density matrix and‖o‖1 =Tr|o| stands for the Schatten1-norm (the so called ‘tracenorm’) of the hermitian operatoro. The quantityN () isequal to |∑i λi|, the modulus of the sum of the negativeeigenvalues of , quantifying the extent to which fails tobe positive. Strictly related toN is the logarithmic negativityEN , defined asEN ≡ ln ‖ ˜‖1, which constitutes an upperbound to thedistillable entanglementof the quantum state.

The negativity has been proven to be convex and monotoneunder LOCC (local operations and classical communications)[40], but fails to be continuous in trace norm on infinite di-mensional Hilbert spaces. However, this problem can be cir-cumvented by restricting to physical states with finite meanenergy [14].

We will now show that for any two–mode Gaussian state the negativity is a simple decreasing function ofν−, whichis thus itself an entanglement monotone. The norm‖·‖1 isunitarily invariant; in particular, it is invariant under globalsymplectic operations in phase space. Considering the sym-plectic diagonalizationν ≡ diag (ν−, ν−, ν+, ν+) of σ, thismeans thatN () = (‖ν‖1 − 1)/2. Now, because of the mul-tiplicativity of the norm‖ · ‖1, we have just to compute thetrace of the single–mode thermal–like operatorρ⊗

νi

ρ⊗

νi=

2

νi + 1

∞∑

k=0

(

νi − 1

νi + 1

)k

|k〉〈k| .

Such an operator is obviously normalized forνi ≥ 1, yielding‖ρ⊗

νi‖1 = 1 (all eigenvalues are positive). Instead, ifνi < 1,

then‖ρ⊗

νi‖1 = 1/νi. The proof is completed by showing that

the largest symplectic eigenvalueν+ of σ fulfills ν+ ≥ 1.This is obviously true for separable states, therefore we cansetc+c− < 0. With this position, it is easy to show that

ν+ > ν− ≥ 1 .

Thus, we obtain‖ρ⊗

ν+‖1 = 1 and

‖ ˜‖1 =1

ν−⇒ N () = max

[

0,1− ν−2ν−

]

, (41)

EN () = max [0,− ln ν−] . (42)

This is a decreasing function of the smallest partially trans-posed symplectic eigenvalueν−, quantifying the amount bywhich inequality (36) is violated. The eigenvalueν− thuscompletely qualifies and quantifies the quantum entanglementof a Gaussian stateσ. Notice that, in such an instance, thisfeature holds for nonsymmetric states as well.

We finally mention that, as far as symmetric states are con-cerned, another measure of entanglement, the entanglementof formationEF [41], can be actually computed [28]. SinceEF turns out to be, again, a decreasing function ofν−, it pro-vides for symmetric states a quantification of entanglementfully equivalent to the one provided by the logarithmic nega-tivity EN .

7

V. SYMPLECTIC EIGENVALUES AND EPRCORRELATIONS

A deeper insight on the relationship between correlationsand the eigenvalueν− is provided by the following observa-tion, which holds in the symmetric case.

Let us define the EPR correlationξ [28, 29] of a continuousvariable two–mode quantum state as

ξ ≡ δx1−x2+ δp1+p2

2=

Trσ

2− σ13 + σ24 , (43)

whereδo = 〈o2〉 − 〈o〉2 for an operatoro. If ξ ≥ 1 then thestate does not possess nonlocal correlations . The idealizedEPR-like state [30] (simultaneous eigenstate of the commut-ing observablesx1 − x2 and p1 + p2) hasξ = 0. As forstandard form states, one has

δx1−x2= a+ b− 2c+ , (44)

δp1+p2= a+ b+ 2c− , (45)

ξ = a+ b− c+ + c− . (46)

Notice thatξ is not by itself a good measure of correlation be-casue, as one can easily verify, it is not invariant under localsymplectic operations. In particular, applying local squeez-ings with parametersri = ln vi and local rotations with anglesϕi to a standard form state, we obtain

ξvi,ϑ =a

2(v21 +

1

v21)+

b

2(v22 +

1

v22)− (c+v1v2−

c−v1v2

) cosϑ ,

(47)with ϑ = ϕ1 + ϕ2. Now, the quantity

ξ ≡ minvi,ϑ

ξvi,ϑ

has to beSp(2,R) ⊕ Sp(2,R) invariant. It corresponds tothe maximal amount of EPR correlations which can be dis-tributed in a two-mode Gaussian state by means of local op-erations. Minimization in terms ofϑ is immediate, yieldingξ = minvi ξvi , with

ξvi =a

2(v21 +

1

v21) +

b

2(v22 +

1

v22)− |c+v1v2 −

c−v1v2

| . (48)

The gradient of such a quantity is null if and only if

a

(

v21 −1

v21

)

− |c+|v1v2 −|c−|v1v2

= 0 , (49)

b

(

v22 −1

v22

)

− |c+|v1v2 −|c−|v1v2

= 0 , (50)

where we introduced the positionc+c− < 0, necessary tohave entanglement. Eqs. (49, 50) can be combined to get

a

(

v21 −1

v21

)

= b

(

v22 −1

v22

)

. (51)

Restricting to the symmetric (a = b) entangled (⇒ c+c− < 0)case, Eq. (51) and the fact thatvi > 0 imply v1 = v2. Under

such a constraint, minimizingξvi becomes a trivial matter andyields

ξ = 2√

(a− |c+|)(a− |c−|) = 2ν− . (52)

We thus see that the smallest symplectic eigenvalue of the par-tially transposed state is endowed with a direct physical inter-pretation: it quantifies the greatest amount of EPR correla-tions which can be created in a Gaussian state by means oflocal operations.

As can be easily shown by a numerical analysis, such a sim-ple interpretation is lost for nonsymmetric states. This factproperly exemplifies the difficulties of handling optimizationproblems in nonsymmetric instances, encountered,e.g.in thecomputation of the entanglement of formation of such states.

VI. PARAMETRIZATION OF GAUSSIAN STATES WITHSYMPLECTIC INVARIANTS

Two–mode Gaussian states can be classified according tothe values of their fourSp(2,R) ⊕ Sp(2,R) invariantsa, b,c+ and c−, which determine their standard form. It is rel-evant to provide a reparametrization of standard form statesin terms of invariants which admit a direct interpretation forgeneric Gaussian states. Such invariants will be the globalpu-rity µ = Tr 2, the marginal purities of the reduced statesµi = Tri[( Trj 6=i)

2] and theSp(4,R) invariant∆, whosemeaning will become soon clear. For a two-mode Gaussianstate one has

µ1 =1

a, µ2 =

1

b, (53)

1

µ2= Detσ = (ab)2 − ab(c2+ + c2−) + (c+c−)

2 , (54)

∆ = a2 + b2 + 2c+c− . (55)

Eqs. (53-55) can be inverted to provide the followingparametrization

a =1

µ1, b =

1

µ2, (56)

c± =1

4

√

√

√

√µ1µ2

[

(

∆− (µ1 − µ2)2

µ21µ

22

)2

− 4

µ2

]

± ǫ

(57)

with ǫ ≡ 1

4

√

[(µ1 + µ2)2 − µ21µ

22∆]

2

µ31µ

32

− 4µ1µ2

µ2.

The global and marginal purities range from0 to 1, con-strained by the condition

µ ≥ µ1µ2 , (58)

that can be easily shown to be a direct consequence of inequal-ity (11). Notice that inequality (58) entails that no Gaussian

8

LPTP (less pure than product) states exist, at variance withthecase of two–qubit systems [6].

The smallest symplectic eigenvalues of the covariance ma-trix σ and of its partial transposeσ are promptly determinedin terms of symplectic invariants

2ν2− = ∆−√

∆2 − 4

µ2, 2ν2− = ∆−

√

∆2 − 4

µ2, (59)

where∆ = −∆+ 2/µ21 + 2/µ2

2.The parametrization provided by Eqs. (56, 57) describes

physical states if the radicals in Eqs. (57, 59) exist and theHeisenberg uncertainty principle (7) is satisfied. All theseconditions can be combined and recast as upper and lowerbounds on the global symplectic invariant∆

2

µ+

(µ1 − µ2)2

µ21µ

22

≤ ∆

≤ min

(µ1 + µ2)2

µ21µ

22

− 2

µ, 1 +

1

µ2

. (60)

Let us investigate the role played by the invariant∆ in thecharacterization of the properties of Gaussian states. To thisaim, we just analyse the dependence of the eigenvalueν− on∆

∂ ν2−∂ ∆

∣

∣

∣

∣

µ1, µ2, µ

=1

2

∆√

∆2 − 14µ2

− 1

> 0 . (61)

The smallest symplectic eigenvalue of the partially transposedstate is strictly monotone in∆. Therefore the entanglement ofa generic Gaussian stateσ with global purityµ and marginalpuritiesµ1,2 strictly increases with decreasing∆. The invari-ant∆ is thus endowed with a direct physical interpretation: atgiven global and marginal purities, it determines the amountof entanglement of the state.

Because, due to inequality (60),∆ possess both lower andupper bounds, not only maximally but alsominimallyentan-gled Gaussian states exist. This elucidates the relations be-tween the entanglement and the purity of two–mode Gaussianstates: the entanglement of such states is tightly bound by theamount of global and marginal purities, with only one remain-ing degree of freedom related to the invariant∆.

VII. EXTREMAL ENTANGLEMENT AT FIXED GLOBALAND LOCAL PURITIES

We now aim to characterize extremal (maximally and mini-mally) entangled Gaussian states for fixed global and marginalpurities. As it is clear from Eq. (53), the standard form ofstates with fixed marginal purities always satisfiesa = 1/µ1,b = 1/µ2. Therefore the complete characterization of maxi-mally and minimally entangled states is achieved by specify-ing the expression of their standard form coefficientsc∓.

Let us first consider the states saturating the lower boundin Eq. (60), which entailsmaximalentanglement. They are

Gaussian maximally entangled states for fixed global and lo-cal purities (GMEMS), admitting the following parametriza-tion

c± = ±√

1

µ1µ2− 1

µ. (62)

It is easily seen that such states belong to the class of asym-metric two–mode squeezed thermal states, with squeezing pa-rameter and symplectic spectrum given by

tanh 2r = 2(µ1µ2 − µ21µ

22/µ)

1/2/(µ1 + µ2) , (63)

ν2∓ =1

µ+

(µ1 − µ2)2

2µ21µ

22

∓ |µ1 − µ2|2µ1µ2

√

(µ1 − µ2)2

µ21µ

22

+4

µ.

(64)In particular, any GMEMS can be written as an entangled two-mode squeezed thermal states [satisfying Ineq. (39)]. Thisprovides a characterization of two-mode thermal squeezedstates as maximally entangled states for given global andmarginal purities. We can restate this result as follows: givenan initial tensor product of (generally different) thermalstates,the unitary operation providing the maximal entanglement forgiven values of the local puritiesµi’s is given by a two-modesqueezing, with squeezing parameter determined by Eq. (63).Notice that fixing the values of the local purities is necessaryto attain maximal, finite entanglment; in fact, fixing only thevalue of the global purityµ always allows for an arbitrarylarge logarithmic negativity (asν− goes to0), achievable bymeans of two–mode squeezing with arbitrary large squeezingparameter.

Nonsymmetric two–mode thermal squeezed states turn outto beseparablein the range

µ ≤ µ1µ2

µ1 + µ2 − µ1µ2. (65)

In such aseparable regionin the space of purities, no entan-glement can occur for states of the form of Eq. (62), while,outside this region, they are properly GMEMS. We remarkthat, as a consequence, all Gaussian states whose purities fallin the separable region defined by inequality (65) are separa-ble.

We now consider the states that saturate the upper bound inEq. (60). They determine the class of Gaussian least entangledstates for given global and local purities (GLEMS). Violationof inequality (65) implies that

1 +1

µ2≤ (µ1 + µ2)

2

µ21µ

22

− 2

µ.

Therefore, outside the separable region, GLEMS fulfill

∆ = 1 +1

µ2. (66)

Eq. (66) expresses partial saturation of Heisenberg relationEq. (12). Namely, considering the symplectic diagonalizationof Gaussian states and Eq. (14), it immediately follows that

9

FIG. 3: Plot of the maximal logarithmic negativity of Gaussian statesfor fixed marginal purities. The surface represents GMEMMS,statesthat saturate inequality (70).

theSp(4,R) invariant condition (66) is fulfilled if and only ifthe symplectic spectrum of the state takes the formν− = 1,ν+ = 1/µ. We thus find that GLEMS are characterized by apeculiar spectrum, with all the mixedness concentrated in one‘decoupled’ quadrature. Moreover, by Eqs. (66) and (12) itfollows that GLEMS are minimum–uncertainty mixed Gaus-sian states. They are determined by the standard form corre-lation coefficients

c± =1

4

√

√

√

√µ1µ2

[

− 4

µ2+

(

1 +1

µ2− (µ1 − µ2)2

µ21µ

22

)2]

± 1

4µ

√

√

√

√

−4µ1µ2 +

[

(1 + µ2)µ21µ

22 − µ2(µ1 + µ2)

2]2

µ2µ31µ

32

.

(67)

Quite remarkably, following the analysis presented in Sec.III,it turns out that the GLEMS at fixed global and marginal pu-rities are also states of minimal globalp−entropy forp < 2,and of maximal globalp−entropy forp > 2.

According to the PPT criterion, GLEMS are separable onlyif µ ≤ µ1µ2/

√

µ21 + µ2

2 − µ21µ

22. Therefore, in the range

µ1µ2

µ1 + µ2 − µ1µ2< µ ≤ µ1µ2

√

µ21 + µ2

2 − µ21µ

22

(68)

both separable and entangled states can be found. Instead, theregion

µ >µ1µ2

√

µ21 + µ2

2 − µ21µ

22

(69)

can only accomodateentangledstates.The very narrow region defined by inequality (68) is thus

the only region of coexistence of both entangled and sepa-rable Gaussian mixed states. We mention that the sufficientcondition for entanglement (69), first derived in Ref. [13] hasbeen independently rediscovered in another recent work [17].

The upper and lower inequalities (60) lead to the followingfurther constraint between the global and the local purities

µ ≤ µ1µ2

µ1µ2 + |µ1 − µ2|. (70)

DEGREES OFPURITY SEPARABILITY

µ < µ1µ2 unphysical region

µ1µ2 ≤ µ ≤ µ1µ2

µ1+µ2−µ1µ2separablestates

µ1µ2

µ1+µ2−µ1µ2< µ ≤ µ1µ2√

µ21+µ2

2−µ2

1µ22

coexistenceregion

µ1µ2√µ21+µ2

2−µ2

1µ22

< µ ≤ µ1µ2

µ1µ2+|µ1−µ2|entangledstates

µ > µ1µ2

µ1µ2+|µ1−µ2|unphysical region

TABLE I: Classification of two–mode Gaussian states and of theirproperties of separability according to their degrees of global purityµ and of marginal puritiesµ1 andµ2.

For purities which saturate inequality (70), one obtains thestates of maximal global purity with given marginal purities.In such an instance GMEMS and GLEMS coincide and wehave a unique class of entangled states depending only on themarginal puritiesµ1,2. They are Gaussian maximally entan-gled states for fixed marginals (GMEMMS). In Fig. 3 the log-arithmic negativity of GMEMMS is plotted as a function ofµ1,2, showing how the maximal entanglement achievable byGaussian states rapidly decreases with increasing differenceof marginal purities, in analogy with finite-dimensional sys-tems [6]. For symmetric states(µ1 = µ2) inequality (70)reduces to the trivial boundµ ≤ 1 and GMEMMS reduce topure two–mode squeezed vacua.

The previous necessary or sufficient conditions for entan-glement are collected in Table I and allow a graphical displayof the behavior of the entanglement of mixed Gaussian statesas shown in Fig. 4. These relations classify the properties ofseparability of all two-mode Gaussian states according to theirdegree of global and marginal purities.

A. Realization of extremally entangled states in experimentalsettings

As we have seen, GMEMS are two-mode squeezed ther-mal states, whose general covariance matrix is described byEqs. (9) and (15). A realistic instance giving rise to such statesis provided by the dissipative evolution of an initially puretwo-mode squeezed vacuum created,e.g., in a non degener-ate parametric down conversion process. Let us denote byσr

the covariance matrix of a two mode squeezed vacuum withsqueezing parameterr, derived from Eqs. (15) withν∓ = 1.The interaction of this initial state with a thermal noise resultsin the following dynamical map describing the time evolutionof the covariance matrixσ(t) [31]

σ(t) = e−Γtσr + (1 − e−Γt)σn1,n2, (71)

whereΓ is the coupling to the noisy reservoir (equal to theinverse of the damping time) andσn1,n2

= ⊕2i=1ni12 is the

covariance matrix of the thermal noise. The average numberof thermal photonsni given by

ni =1

exp (~ωi/kBT )− 1

10

FIG. 4: Summary of entanglement properties of two–mode (nonsymmetric) Gaussian states in the space of marginal purities µ1,2 (x- andy-axes) and global purityµ. In fact, on thez-axis we plot the ratioµ/µ1µ2 to gain a better graphical distinction between the various regions.In this space, all physical states lay between the horizontal planez = 1 representing product states, and the upper limiting surface representingGMEMMS. Separable and entangled states are well separated except for a narrow region of coexistence (depicted in black). Separable statesfill the region depicted in dark grey, while in the region containing only entangled states we have depicted the average logarithmic negativityEq. (88), growing from white to medium grey. The mathematical relations defining the boundaries between all these regions are collected inTable I. The three-dimensional envelope is cut atz = 3.5.

in terms of the frequencies of the modesωi and of the temper-ature of the reservoirT . It can be easily verified that the co-variance matrix Eq. (71) defines a two-mode thermal squeezedstate, generally nonsymmetric (forn1 6= n2). However, no-tice that the entanglement of such a state cannot persist in-definetely, becaus after a given time inequality (39) will beviolated and the state will evolve into a non entangled two-mode squeezed thermal state. We also notice that the relevantinstance of pure loss (n1 = n2 = 0) allows the realization ofsymmetric GMEMS.

Concerning the experimental characterization of minimallyentangled states, one can envisage several explicit experimen-tal settings for their realiazion. For instance, let us consider(see Fig. 5) a beam splitter with transmittivityτ = 1/2 (corre-sponding to a two-mode rotation of angleπ/4 in phase space).Suppose that a single-mode squeezed state, with covariancematrixσ1r = diag ( e2r, e−2r) (like, e.g., the result of a de-generate parametric down conversion in a nonlinear crystal),enters in the first input of the beam splitter. Let the other in-put be an incoherent thermal state produced from a source atequilibrium at a temperatureT . The purityµ of such a statecan be easily computed in terms of the temperatureT and ofthe frequency of the thermal modeω

µ =exp (~ω/kBT )− 1

exp (~ω/kBT ) + 1. (72)

The state at the output of the beam splitter will be a corre-lated two-mode Gaussian state with covariance matrixσout

that reads

σout =1

2

n+ k 0 n− k 0

0 n+ k−1 0 n− k−1

n− k 0 n+ k 0

0 n− k−1 0 n+ k−1

,

one-mode squeezed beam

one-mode thermal state

BS 50:50

OPO / OPA

two-

mod

e

GLEM

S

FIG. 5: Possible scheme for the generation of Gaussian leasten-tangled mixed states (GLEMS). A single-mode squeezed state(ob-tained, for example, by an optical parametric oscillator oramplifier)interferes with a thermal state through a 50:50 beam splitter. Theresulting two-mode state is a minimally entangled mixed Gaussianstate at given global and marginal purities.

11

with k = e2r andn = µ−1. By immediate inspection, thesymplectic spectrum of this covariance matrix isν− = 1 andν+ = 1/µ. Therefore the output state is always a state withextremal generalized entropy at a given purity (the state canbe seen as the tensor product of a vacuum and of a thermalstate, on which one has applied symplectic transformation).Moreover, the state is entangled if

cosh(2r) >µ2 + 1

2µ=

exp (2~ω/kBT ) + 1

exp (2~ω/kBT )− 1. (73)

Tuning the experimental parameters to meet the above con-dition, indeed makes the output state of the beam splitter asymmetric GLEMS. It is interesting to observe that nonsysm-metric GLEMS can be produced as well by choosing a beamsplitter with transmissivity different from0.5.

VIII. EXTREMAL ENTANGLEMENT AT FIXED GLOBALAND LOCAL GENERALIZED ENTROPIES

In this section we introduce a more general characterizationof the entanglement of generic two–mode Gaussian states, byexploiting the generalizedp−entropies as measures of globaland marginal mixedness. For ease of comparison we willcarry out this analysis along the same lines followed in theprevious Section, by studying the explicit behavior of theglobal invariant∆, directly related to the logarithmic nega-tivity EN at fixed global and marginal entropies. This studywill clarify the relation between∆ and the generalized en-tropiesSp and the ensuing consequences for the entanglementof Gaussian states.

We begin by observing that the standard form covari-ance matrixσ of a generic two–mode Gaussian state can beparametrized by the following quantities: the two marginalsµ1,2 (or any other marginalSp1,2

because all the local, single-mode entropies are equivalent for any value of the integerp),the globalp−entropySp (for some chosen value of the integerp), and the global symplectic invariant∆. On the other hand,Eqs. (18,21,13) provide an explicit expression for anySp as afunction ofµ and∆. Such an expression can be exploited tostudy the behaviour of∆ as a function of the global purityµ,at fixed marginals and globalSp (from now on we will omitthe explicit reference to fixed marginals). One has

∂µ

∂∆

∣

∣

∣

∣

Sp

= − 2

R2

∂R

∂∆

∣

∣

∣

∣

Sp

=2

R2

∂Sp/∂∆|R∂Sp/∂R|∆

=2

R2

Np(∆, R)

Dp(∆, R), (74)

where we have defined the inverse participation ratio

R ≡ 2

µ, (75)

and the remaining quantitiesNp andDp read

Np(∆, R) =[

(R + 2 + 2√∆+R)p−1

− (R + 2− 2√∆+R)p−1

]√∆−R

−[

(R − 2 + 2√∆−R)p−1

− (R − 2− 2√∆−R)p−1

]√∆+R ,

Dp(∆, R) =[

(√∆+ R+ 1)(R+ 2 + 2

√∆+R)p−1

+ (√∆+R− 1)(R + 2− 2

√∆+R)p−1

]

·√∆−R

−[

(√∆− R+ 1)(R− 2− 2

√∆−R)p−1

+ (√∆−R− 1)(R − 2 + 2

√∆−R)p−1

]

·√∆+R . (76)

Now, it is easily shown that the ratioNp(∆, R)/Dp(∆, R) isincreasing with increasingp and has a zero atp = 2 for any∆, R; in particular, its absolute minimum(−1) is reached inthe limit (∆ → 2, R → 2, p → 1). Thus the derivativeEq. (74) is negative forp < 2, null for p = 2 (in this case∆andS2 = 1 − µ are of course regarded as independent vari-ables) and positive forp > 2. This implies that, for givenmarginals, keeping fixed any globalSp for p < 2 the min-imum (maximum) value of∆ corresponds to the maximum(minimum) value of the global purityµ. Instead, by keepingfixed any globalSp for p > 2 the minimum of∆ is alwaysattained at the minimum of the global purityµ.

This observation allows to determine rather straightfor-wardly the states with extremal∆. They are extremally entan-gled states because, for fixed global and marginal entropies,the logarithmic negativity of a state is determined only by theone remaining independent global symplectic invariant, rep-resented by∆ in our choice of parametrization. If, for themoment being, we neglect the fixed local purities, then thestates with maximal∆ are the states with minimal (maximal)µ for a given globalSp with p < 2 (p > 2) (see Sec. IIIand Fig. 1). As found in Sec. III, such states are minimumuncertainty two–mode states with mixedness concentrated inone quadrature. We have shown in Sec. VII that they cor-respond to Gaussian least entangled mixed states (GLEMS)whose standard form is given by Eq. (67). As can be seenfrom Eq. (67), these states are consistent with any legitimatephysical value of the local invariantsµ1,2. We therefore con-clude thatall Gaussian states withmaximal∆ for any fixedtriple of values of global and marginal entropies are GLEMS.

Viceversa one can show thatall Gaussian states withmin-imal ∆ for any fixed triple of values of global and marginalentropies are Gaussian maximally entangled mixed states(GMEMS). This fact is immediately evident in the symmetriccase because the extremal surface in theSp vs.SL diagrams isalways represented by symmetric two–mode squeezed states(symmetric GMEMS). These states are characterized by adegenerate symplectic spectrum and encompass only equal

12

choices of the local invariants:µ1 = µ2. In the nonsymmetriccase, the given values of the local entropies are different,andthe extremal value of∆ is further constrained by inequality(60)

∆−R ≥ (µ1 − µ2)2

µ21µ

22

. (77)

From Eq. (74) it follows that

∂(∆−R)

∂∆

∣

∣

∣

∣

Sp,µ1,2

= 1 +Np(∆, R)

Dp(∆, R)≥ 0 , (78)

becauseNp(∆, R)/Dp(∆, R) > −1. Thus,∆ − R is an in-creasing function of∆ at fixedµ1,2 andSp, and the minimal∆ corresponds to the minimum of∆ − R, which occurs ifinequality (77) is saturated. Therefore, also in the nonsym-metric case, the two-mode Gaussian states with minimal∆ atfixed global and marginal entropies are GMEMS.

Summing up, we have shown that GMEMS and GLEMS in-troduced in the previous section at fixed global and marginallinear entropies, are alwaysextremallyentangled Gaussianstates, whatever triple of generalized global and marginalen-tropic measures one chooses to fix. Maximally and minimallyentangled states of CV systems are thus very robust with re-spect to the choice of different measures of mixedness. Thisis at striking variance with the case of discrete variable sys-tems, where it has been shown that fixing different measuresof mixedness yields different classes of maximally entangledstates [4].

Furthermore, we will now show that the characterizationprovided by the generalized entropies leads to some remark-able new insight on the behavior of the entanglement of CVsystems. The crucial observation is that for a genericp, thesmallest symplectic eigenvalue of the partially transposed co-variance matrix, at fixed global and marginalp−entropies, isnot in general a monotone function of∆, so that the connec-tion between extremal∆ and extremal entanglement turns outto be, in some cases, inverted. In particular, while forp < 2the GMEMS and GLEMS surfaces tend to be more separatedasp decreases, forp > 2 the two classes of extremally en-tangled states get closer with increasingp and, within a par-ticular range of global and marginal entropies, they exchangetheir role. GMEMS (i.e.states with minimal∆) become min-imally entangled states and GLEMS (i.e.states with maximal∆) become maximally entangled states. This inversion alwaysoccurs for allp > 2.

To understand this interesting behaviour, let us study thedependence of the symplectic eigenvalueν− on the global in-variant∆ at fixed marginals and at fixedSp for a genericp.Using Maxwell’s relations, we can write

κp ≡ ∂(2ν2−)

∂∆

∣

∣

∣

∣

Sp

=∂(2ν2−)

∂∆

∣

∣

∣

∣

R

− ∂(2ν2−)

∂R

∣

∣

∣

∣

∆

· ∂Sp/∂∆|R∂Sp/∂R|∆

.

(79)Clearly, forκp > 0 GMEMS and GLEMS retain their usualinterpretation, whereas forκp < 0 they exchange their role.On thenodeκp = 0 GMEMS and GLEMS share the sameentanglement,i.e. the entanglement of all Gaussian states

(a)

(b)

FIG. 6: Plot of the nodal surface which solves the equationκp = 0with κp defined by Eq. (80), for (a)p = 3 and (b)p = 4. Theentanglement of Gaussian states that lie on the leaf–shapedsurfacesis fully quantified in terms of the marginal purities and the globalgeneralized entropy (a)S3 or (b)S4. The equations of the surfacesin the spaceEp ≡ µ1, µ2, Sp are given by Eqs. (83–85).

at κp = 0 is fully determined by the global and marginalp−entropies alone, and does not depend any more on∆. Suchnodes also exist in the casep ≤ 2 in two limiting instances:in the special case of GMEMMS (states with maximal globalpurity at fixed marginals) and in the limit of zero marginalpurities. We will now show that, besides these two limitingbehaviors, a nontrivial node appears for allp > 2, implyingthat on the two sides of the node GMEMS and GLEMS in-deed exhibit opposite behaviors. Because of Eq. (74),κp canbe written in the following form

κp = κ2 −R

√

∆2 −R2

Np(∆, R)

Dp(∆, R), (80)

with Np andDp defined by Eq. (76) and

∆ = −∆+2

µ21

+2

µ22

,

κ2 = −1 +∆

√

∆2 −R2,

The quantityκp in Eq. (80) is a function ofp, R, ∆, andof the marginals; since we are looking for the node (wherethe entanglement is independent of∆), we can investigate theexistence of a nontrivial solution to the equationκp = 0 fixingany value of∆. Let us choose∆ = 1+R2/4 that saturates theHeisenberg uncertainty relation and is satisfied by GLEMS.

13

With this position, Eq. (80) becomes

κp(µ1, µ2, R) = κ2(µ1, µ2, R)

− R√

(

2µ21

+ 2µ22

− R2

4 − 1)2

−R2

fp(R) .

(81)

The existence of the node depends then on the behavior of thefunction

fp(R) ≡ 2[

(R + 2)p−2 − (R − 2)p−2]

(R + 4)(R+ 2)p−2 − (R− 4)(R− 2)p−2. (82)

In fact, as we have already pointed out,κ2 is always posi-tive, while the functionfp(R) is an increasing function ofpand, in particular, it is negative forp < 2, null for p = 2and positive forp > 2, reaching its asymptote2/(R + 4) inthe limit p → ∞. This entails that, forp ≤ 2, κp is alwayspositive, which in turn implies that GMEMS and GLEMS arerespectively maximally and minimally entangled two–modestates with fixed marginal and globalp−entropies in the rangep ≤ 2 (including both von Neumann and linear entropies).On the other hand, for anyp > 2 one node can be foundsolving the equationκp(µ1, µ2, 2/µ) = 0. The solutionsto this equation can be found analytically for lowp and nu-merically for anyp. They form a continuum in the spaceµ1, µ2, µ which can be expressed as a surface of generalequationµ = µκ

p(µ1, µ2). Since the fixed variable isSp andnot µ it is convenient to rewrite the equation of this surfacein the spaceEp ≡ µ1, µ2, Sp, keeping in mind the rela-tion (34), holding for GLEMS, betweenµ andSp. In this waythe nodal surface(κp = 0) can be written in the form

Sp = Sκp (µ1, µ2) ≡

1− gp[

(µκp(µ1, µ2))

−1]

p− 1. (83)

The entanglement of all Gaussian states whose entropies lieon the surfaceSκ

p (µ1, µ2) is completelydeterminated by theknowledge ofµ1, µ2 andSp. The explicit expression of thefunctionµκ

p(µ1, µ2) depends onp but, being the global purityof physical states, is constrained by the inequality

µ1µ2 ≤ µκp(µ1, µ2) ≤ µ1µ2

µ1µ2 + |µ1 − µ2|.

The nodal surface of Eq. (83) constitutes a ‘leaf’, with baseatthe pointµκ

p(0, 0) = 0 and tip at the pointµκp(√3/2,

√3/2) =

1, for anyp > 2; such a leaf becomes larger and flatter withincreasingp (see Fig. 6). Forp > 2, the functionfp(R) de-fined by Eq. (82) is negative but decreasing with increasingR,that is with decreasingµ. This means that, in the space of en-tropiesEp, above the leaf(Sp > Sκ

p ) GMEMS (GLEMS) arestill maximally (minimally) entangled states for fixed globaland marginal generalized entropies, while below the leaf theyare inverted. Notice also that forµ1,2 >

√3/2 no node and

so no inversion can occur for anyp. Each point on the leaf–shaped surface of Eq. (83) corresponds to an entire class of in-finitely many two–mode Gaussian states (including GMEMS

and GLEMS) with the same marginals and the same globalSp = Sκ

p (µ1, µ2), which are all equally entangled, since theirlogarithmic negativity is completely determined byµ1, µ2 andSp. For the sake of clarity we provide the explicit expressionsof µκ

p(µ1, µ2), as plotted in Fig. 6 for the cases (a)p = 3, and(b) p = 4.

µκ3 (µ1, µ2) =

(

63µ21

+ 3µ21

− 2

)12

, (84)

µκ4 (µ1, µ2) =

√3µ1µ2

/ (

µ21 + µ2

2 − 2µ21µ

22 +

√

(µ21 + µ2

2) (µ21 + µ2

2 − µ21µ

22) + µ4

1µ42

)12

.

(85)

Apart from the relevant ‘inversion’ feature shown byp−entropies forp > 2, the possibility of an accurate charac-terization of CV entanglement based on global and marginalentropic measures still holds in the general case for anyp. Inparticular, the set of all Gaussian states can be again divided,in the space of global and marginalSp’s, into three main ar-eas: separable, entangled and coexistence region. It can bethus very interesting to investigate how the different entropicmeasures chosen to quantify the degree of global mixedness(all marginal measures are equivalent) behave in classifyingthe separability properties of Gaussian states. Fig. 7 providesa numerical comparison of the different characterizationsofentanglement obtained by the use of differentp−entropies,with p ranging from1 to 4, for symmetric Gaussian states(Sp1

= Sp2≡ Spi

). The last restriction has been imposed justfor ease of graphical display. The following considerations,based on the exact numerical solutions of the transcendentalconditions, will take into account nonsymmetric states as well.

The mathematical relations expressing the boundaries be-tween the different regions in Fig. 7 are easily obtained foranyp by starting from the relations holding forp = 2 (see Ta-ble I) and by evaluating the correspondingSp(µ1,2) for eachµ(µ1,2). For any physical symmetric state such a calculationyields

0 ≤ (p− 1)Sp < 1− gp

(

√

2− µ2i

µi

)

⇒ entangled,

1− gp

(

√

2− µ2i

µi

)

≤ (p− 1)Sp < 1− g2p

(√

2− µi

µi

)

⇒ coexistence,

1− g2p

(√

2− µi

µi

)

≤ (p− 1)Sp ≤ 1− g2p

(

1

µ2i

)

⇒ separable. (86)

Equations (86) were obtained exploiting the multiplicativityof p−norms on product states and using Eq. (27) for the lower

14

(a) (b)

(c) (d)

FIG. 7: Summary of the entanglement properties for symmetric Gaussian states at fixed global and marginal generalizedp−entropies, for (a)p = 1 (von Neumann entropies), (b)p = 2 (linear entropies), (c)p = 3, and (d)p = 4. In the entangled region, the average logarithmicnegativityEN (Spi , Sp) Eq. (88) is depicted, growing from gray to black. Forp > 2 an additional dashed curve is plotted; it represents thenodal line of inversion. Along it the entanglement is fully determined by the knowledge of the global and marginal generalized entropiesSpi , Sp, and GMEMS and GLEMS are equally entangled. On the left side of the nodal line GMEMS (GLEMS) are maximally (minimally)entangled Gaussian states at fixedSpi , Sp. On the right side of the nodal line they are inverted: GMEMS (GLEMS) are minimally (maximally)entangled states. Also notice how the dashed region of coexistence becomes narrower with increasingp. The equations of all boundary curvescan be found in Eq. (86).

boundary of the coexistence region (which represents GLEMSbecoming entangled) and Eq. (30) for the upper one (whichexpresses GMEMS becoming separable). Let us mention alsothat the relation between any local entropic measureSpi

andthe local purityµi is obtained directly from Eq. (21) and reads

Spi=

1− gp(1/µi)

p− 1. (87)

We noticeprima faciethat, with increasingp, the entan-glement is more sharply qualified in terms of the global andmarginalp−entropies. In fact the region of coexistence be-

15

tween separable and entangled states becomes narrower withhigherp. Thus, somehow paradoxically, with increasingp theentropySp provides less information about a quantum state,but at the same time it yields a more accurate characterizationand quantification of its entanglement. In the limitp → ∞all the physical states collapse to one point at the origin oftheaxes in the space of generalized entropies, due to the fact thatthe measureS∞ is identically zero.

IX. QUANTIFYING ENTANGLEMENT: THE AVERAGELOGARITHMIC NEGATIVITY

We have extensively shown that knowledge of the globaland marginal generalizedp−entropies accurately character-izes the entanglement of Gaussian states, providing strongsufficient or necessary conditions. The present analysis nat-urally leads us to propose an actualquantificationof entan-glement, based exclusively on marginal and global entropicmeasures, enriching and generalizing the approach introducedin Ref. [13].

Outside the separable region, we can formally define themaximal entanglementENmax(Sp1,2

, Sp) as the logarithmicnegativity attained by GMEMS (or GLEMS, below the inver-sion nodal surface forp > 2, see Fig. 6). In a similar way, inthe entangled region GLEMS (or GMEMS, below the inver-sion nodal surface forp > 2) achieve the minimal logarithmicnegativityENmax(Sp1,2

, Sp). The explicit analytical expres-sions of these quantities are unavailable for anyp 6= 2 dueto the transcendence of the conditions relatingSp to the sym-plectic eigenvalues. The surfaces of maximal and minimal en-tanglement in the space of the global and localSp are plottedin Fig. 8 for symmetric states. In the planeSp = 0 the up-per and lower bounds correctly coincide, since for pure statesthe entanglement is completely quantified by the marginal en-tropy. For mixed states this is not the case but, as the plotshows, knowledge of the global and marginal entropies strictlybounds the entanglement both from above and from below.For p > 2, we notice how GMEMS and GLEMS exchangetheir role beyond a specific curve in the space ofSp’s. Theequation of this nodal curve is obtained from the general leaf–shaped nodal surfaces of Eqs. (83–85), by imposing the sym-metry constraint (Sp1

= Sp2≡ Spi

). We notice again how theSp’s with higherp provide a better characterization of the en-tanglement, even quantitatively. In fact, the gap between thetwo extremally entangled surfaces in theSp’s space becomessmaller with higherp. Of course the gap is exactly zero allalong the nodal line of inversion forp > 2.

We will now introduce a particularly convenient quantita-tive estimate of the entanglement based only on the knowl-edge of the global and marginal entropies. Let us define the“average logarithmic negativity”EN as

EN (Sp1,2, Sp) ≡

ENmax(Sp1,2, Sp) + ENmin(Sp1,2

, Sp)

2.

(88)We will now show that this quantity, fully determined by theglobal and marginal entropies, provides a reliable quantifica-tion of entanglement (logarithmic negativity) for two–mode

Gaussian states. To this aim, we define the relative errorδEN

on EN as

δEN (Sp1,2, Sp) ≡

ENmax(Sp1,2, Sp)− ENmin(Sp1,2

, Sp)

ENmax(Sp1,2, Sp) + ENmin(Sp1,2

, Sp).

(89)As Fig. 9 shows, this error decreasesexponentiallyboth withdecreasing global entropy and increasing marginal entropies,that is with increasing entanglement. In general the relativeerrorδEN is ‘small’ for sufficiently entangled states; we willpresent more precise numerical considerations in the subcasep = 2. Notice that the decaying rate of the relative error isfaster with increasingp: the average logarithmic negativityturns out to be a better estimate of entanglement with increas-ing p. Forp > 2, δEN is exactly zero on the inversion node,then it becomes finite again and, after reaching a local maxi-mum, it goes asymptotically to zero (see the insets of Fig. 9).

All the above considerations, obtained by an exact numeri-cal analysis, show that the average logarithmic negativityEN

at fixed global and marginalp−entropies is a very good es-timate of entanglement in CV systems, whose reliability im-proves with increasing entanglement and, surprisingly, withincreasing orderp of the entropic measures.

A. Direct estimate of the entanglement

In the present general framework, a peculiar role is playedby the casep = 2, i.e. by the linear entropySL (or, equiv-alently, the purityµ). The previous general analysis on thewhole range of generalized entropiesSp, has remarkablystressed the privileged theoretical role of the instancep = 2,which discriminates between the region in which extremallyentangled states are unambiguously characterized and the re-gion in which they can exchange their roles. Moreover, thegraphical analysis shows that, in the region where no inver-sion takes place(p ≤ 2), fixing the globalS2 = 1 − µ yieldsthe most stringent constraints on the logarithmic negativity ofthe states (see Figs. 7, 8, 9). Notice that such constraints,involving no transcendental functions forp = 2, can be eas-ily handled analytically. A crucial experimental considerationstrengthens these theoretical and practical reasons to privilegethe role ofS2. In fact,S2 can indeed, assuming some priorknowledge about the state (essentially, its Gaussian character),be measured through conceivable direct methods, in particularby means of single–photon detection schemes [17] or of theupcoming quantum network architectures [42]. No completehomodyne reconstruction of the density matrix is needed insuch schemes. Very recently, a first important and promisingstep in this direction has been realized with the experimentalimplementation of direct photon detection for the measure-ment of the squeezing and purity of a single-mode squeezedvacuum with a setup that required only a tunable beam splitterand a single-photon detector [43].

As already anticipated, forp = 2 we can provide analyti-cal expressions for the extremal entanglement in the space of

16

(a) (b)

(c) (d)

FIG. 8: Upper and lower bounds on the logarithmic negativityof symmetric Gaussian states as functions of the global and marginal generalizedp−entropies, for (a)p = 1 (von Neumann entropies), (b)p = 2 (linear entropies), (c)p = 3, and (d)p = 4. The black (gray) surface representsGMEMS (GLEMS). Notice that forp > 2 GMEMS and GLEMS surfaces intersect along the inversion line(meaning they are equally entangledalong that line), and beyond it they interchange their role.The equations of the inversion lines are obtained from Eqs. (83–85), with the positionSp1 = Sp2 ≡ Spi .

global and marginal purities

ENmax(µ1,2, µ) = −1

2log

[

− 1

µ+

(

µ1 + µ2

2µ21µ

22

)

×

µ1 + µ2 −√

(µ1 + µ2)2 −4µ2

1µ22

µ

]

, (90)

ENmin(µ1,2, µ) = −1

2log

[

1

µ21

+1

µ22

− 1

2µ2− 1

2

−√

(

1

µ21

+1

µ22

− 1

2µ2− 1

2

)2

− 1

µ2

]

. (91)

Consequently, both the average logarithmic negativityδEN ,defined in Eq. (88), and the relative errorδEN , given byEq. (89), can be easily evaluated in terms of the purities. Therelative error is plotted in Fig. 9(b) for symmetric states as afunction of the ratioSLi

/SL. Notice, as already pointed outin the general instance of arbitraryp, how the error decaysexponentially. In particular, it falls below5% in the rangeSL < SLi

(µ > µi), which excludes at most very weaklyentangled states (states withEN . 1 [44]). Let us remarkthat the accuracy of estimating entanglement by the averagelogarithmic negativity proves even better in the nonsymmetriccaseµ1 6= µ2, essentially because the maximal allowed entan-

17

(a) (b)

(c) (d)

FIG. 9: The relative errorδEN Eq. (89) on the average logarithmic negativity as a functionof the ratioSpi/Sp, for (a)p = 1, (b) p = 2, (c)p = 3, (d) p = 4, plotted at (a)SV = 1, (b)SL = 1/2, (c)S3 = 1/4, (d)S4 = 1/6. Notice how, in general, the error decays exponentially,and in particular faster with increasingp. Forp > 2, notice how the error reaches zero on the inversion node (seethe insets), then grows andreaches a local maximum before going back to zero asymptotically.

glement decreases with the difference between the marginals,as shown in Fig. (3)

The above analysis proves that the average logarithmic neg-ativity EN is a reliable estimate of the logarithmic negativ-ity EN , improving as the entanglement increases. This al-lows for an accurate quantification of continuous variable en-tanglement by knowledge of the global and marginal puri-ties. As we already mentioned, the latter quantities may bein turn amenable to direct experimental determination by ex-ploiting recent single–photon detection proposals or the up-coming technology of quantum networks.

X. SUMMARY AND CONCLUDING REMARKS

Summarizing, we have pointed out the existence of bothmaximally and minimally entangled two–mode Gaussianstates at fixed local and global generalizedp−entropies. Theanalytical properties of such states have been studied in de-tail for any value ofp. Remarkably, forp ≤ 2, minimallyentangled states are minimum uncertainty states, saturatingthe Heisenberg principle, while maximally entangled statesare nonsymmetric two–mode squeezed thermal states. Inter-estingly, forp > 2 and in specific ranges of the values ofthe entropic measures, the role of such states is reversed. Inparticular, for such quantifications of the global and localen-

tropies, two–mode squeezed thermal states, often referredtoas continuous variable analog of maximally entangled states,turn out to beminimallyentangled.

In the search for the extremally entangled states, we fo-cused on the hierarchy of mixedness induced byp−entropieson the set of arbitrary multi–mode states, investigating severalrelated subjects, like the ordering of such different entropicmeasures and the analytical comparison between the genericSp (and in particular the von Neumann entropySV ) and thelinear entropySL. Moreover, we have introduced the no-tion of “average logarithmic negativity” for given global andmarginal generalizedp-entropies, showing that it provides areliable estimate of CV entanglement in a wide range of phys-ical parameters.

Our analysis also clarifies the reasons why the linear en-tropy is a ‘privileged’ measure of mixedness in continuousvariable systems. It is naturally normalized between0 and1, it offers an accurate qualification and quantification of en-tanglement of any mixed state while giving significative in-formation about the state itself and, crucially, is the onlyen-tropic measure which could be directly measured in the nearfuture by schemes involving only single-photon detectionsorthe technology of quantum networks, without requiring a fullhomodyne reconstruction of the state.

More generally, the present analysis shows that some of thecanonical measures of entanglement and mixedness in the dis-

18

crete variable scenario, such as the entanglement of formationand the von Neumann entropy, may not be the best choicesfor the characterization of mixed states of continuous variablesystems. Discontinuous behaviors appear in the limit of in-finite dimensions for quantities like the entanglement of for-mation, even when restricted to the special class of Gaussianstates. Entanglement measures such as the logarithmic nega-tivity, and informational measures such as the linear entropyand the generalized entropies of higher order provide quantifi-

cations which are, in many respects, more satisfactory (bothmathematically and physically) in the context of CV systems.

Acknowledgments

We thank INFM, INFN, and MIUR under national projectPRIN-COFIN 2002 for financial support.

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[44] It is straightforwad to verify that, in the instance of two–modesqueezed thermal (symmetric) states, such a condition corre-sponds tocosh(2r) & µ1/4. This constraint can be easily sat-isfied with the present experimental technology: even for thequite unfavorable caseµ = 0.5 the squeezing parameter neededis justr ≃ 0.3.