We classify the entanglement of two--mode Gaussian states according to their
degree of total and partial mixedness. We derive exact bounds that determine
maximally and minimally entangled states for fixed global and marginal
purities. This characterization allows for an experimentally reliable estimate
of continuous variable entanglement based on measurements of purity.Comment: 4 pages, 3 EPS figures. Final versio

Alessio Serafini

Fabrizio Illuminati

G. Adesso

Gerardo Adesso

English

arXiv

We classify the entanglement of two-mode Gaussian states according to their degree of total and partial mixedness. We derive exact bounds that determine maximally and minimally entangled states for fixed global and marginal purities. This characterization allows for an experimentally reliable estimate of continuous variable entanglement based on measurements of purity

Adesso, G

Serafini, A

Illuminati, F

UCL Discovery

arXiv:quant-ph/0310150v4  25 Feb 2004Determination of continuous variable entanglement by purity measurementsGerardo Adesso, Alessio Seraﬁni, and Fabrizio IlluminatiDipartimento di Fisica “E. R. Caianiello”, Universit` a di Salerno, INFM UdR di Salerno,INFN Sezione di Napoli, Gruppo Collegato di Salerno, Via S. Allende, 84081 Baronissi (SA), Italy(Dated: February 25, 2004)We classify the entanglement of two–mode Gaussian states according to their degree of total and partialmixedness. We derive exact bounds that determine maximally and minimally entangled states for ﬁxed globalandmarginal purities. Thischaracterizationallowsfor anexperimentallyreliableestimateofcontinuous variableentanglement based on measurements of purity.PACS numbers: 03.67.Mn, 03.65.UdQuantumentanglementofGaussianstates constitutesafun-damentalresourceincontinuousvariable(CV)quantuminfor-mation [1]. Therefore, the quest for a theoretically satisfyingand experimentally realizable quantiﬁcation of the entangle-mentforsuch states stands as a majorissue inthe ﬁeld. On thetheoretical ground, a proper, computable quantitative charac-terization of the entanglement of Gaussian states is providedby the logarithmic negativity [2]. Experimental schemes todetermine the entanglement of Gaussian states have been pro-posed both in the two–mode [3] and in the multipartite [4]instance. However, these schemes are based on homodynedetections, and require a full reconstruction of all the secondmoments of the Gaussian ﬁeld.In this work, we present a theoretical framework to esti-mate the entanglement of two–mode Gaussian states by theknowledge of the total and of the two partial purities. Thisis achieved by deriving analytical a priori upper and lowerbounds on the logarithmic negativity for ﬁxed values of theglobal and marginal purities. We then show that the set of en-tangled Gaussian states is tightly contained between two ex-tremal surfaces of maximally and minimally entangled states.This quantiﬁcation allows for a simple strategy to measurethe entanglement of Gaussian states with reliable experimen-tal accuracy. In fact, measurements of global and marginalpurities do not require the demanding reconstruction of thefull covariance matrix and can be performed directly by ex-ploiting the technology of quantum networks [5].Let us consider a two–mode continuous variable system,described by the Hilbert space H = H1 ⊗ H2 resulting fromthe tensor product of the Fock spaces Hk’s. We denote by akthe annihilation operator acting on the space Hk. Likewise,ˆ xk = (ak + a†k)/√2 and ˆ pk = −i(ak − a†k)/√2 are thequadrature phase operators of the mode k, the correspondingphase space variables being xk and pk.In the following, we will make use of the Wigner quasi–probability representation W(xi,pi), deﬁned as the Fouriertransform of the symmetrically ordered characteristic func-tion. In Wigner phase space picture, the tensor product H =H1 ⊗ H2 results in the direct sum Γ = Γ1 ⊕ Γ2 of the relatedphase spaces Γi’s. A symplectic transformation acting on theglobal phase space Γ corresponds to a unitary operator actingon H [6]. We will refer to a transformation Sl = S1 ⊕ S2,with each Si ∈ Sp(2,R) acting on Γi, as to a “local symplecticoperation”, corresponding to a “local unitary transformation”Ul = U1 ⊗U2. The set of Gaussian states is deﬁned as the setof states with Gaussian Wigner functionW(X) =e− 12Xσ−1XTπ√Detσ, (1)where X ≡ (x1,p1,x2,p2) ∈ Γ, and we will denote by ˆ X thevector of operators (ˆ x1, ˆ p1, ˆ x2, ˆ p2). First moments have beenneglected, since they can be set to zero by means of a localunitary transformation. Second moments form the covariancematrix σ of the Gaussian state σij ≡ 12  ˆ Xi ˆ Xj + ˆ Xj ˆ Xi  −  ˆ Xi   ˆ Xj . For simplicity, in what follows σ will refer bothto the Gaussian state and to its covariance matrix. It is conve-nient to express σ in terms of the three 2 × 2 matrices α, β,γσ ≡ α γγT β . (2)Heisenberg uncertainty principle can be expressed as [6]σ +i2Ω ≥ 0 , (3)where Ω ≡ ω ⊕ ω is the usual symplectic form with ωij =δij−1−δij+1, i,j = 1,2. Ineq.(3)canberecastasaconstrainton the Sp(4,R) invariants ∆ ≡ Detα+ Detβ+2Detγ, andDetσ [6]∆ ≤14+ 4Detσ . (4)Ingeneral, theWigner functiontransformsas a scalar undersymplectic operations, while the covariance matrix σ trans-forms according to σ → STσS, with S ∈ Sp(4,R). As itis well known [7], for any covariance matrix σ there exists alocal canonical operation Sl = S1 ⊕ S2 that recasts σ in the“standard form” σsf with α = diag{a,a}, β = diag{b,b},γ = diag{c+,c−}, where a, b, c+, c− are determined bythe four local symplectic invariants Detσ, Detα, Detβ, andDetγ.Any bipartite Gaussian state σ can always be writ-ten as σ = STνS for some S ∈ Sp4,R and ν =diag{n−,n−,n+,n+}. The quantities n∓ constitute the2symplectic spectrum of σ; they are determined by the globalsymplectic invariants [8, 9]2n2∓ = ∆ ∓ ∆2 − 4Detσ . (5)In terms of n∓ Ineq. (4) becomes simply n− ≥ 1/2.We will characterize the mixedness of a quantum state ̺by its purity   ≡ Tr̺2. For a n–mode Gaussian state σthe purity is simply evaluatedintegratingthe Wigner function,yielding   = 1/(2n√Detσ).As for the entanglement, we recall that the positivity of thepartially transposed (PPT) state ˜ σ is equivalent to separabilityfor any two–mode Gaussian state σ [6]. In terms of symplec-tic invariants, partial transposition corresponds to ﬂipping thesign of Detγ, so that ∆ turns into ˜ ∆ = ∆ − 4Detγ. Thesymplectic spectrum ˜ n∓ of ˜ σ is simply found inserting ˜ ∆ for∆ in Eq. (5). If ˜ n− is the smallest symplectic eigenvalue ofthe partially transposed covariance matrix ˜ σ, a state σ is sep-arable if and only if˜ n− ≥ 1/2 . (6)A bona ﬁde measure of entanglement for two–mode Gaus-sian states shouldthus bea monotonicallydecreasingfunctionof ˜ n−, quantifying the violation of inequality (6). A com-putable entanglement monotone for generic two-mode Gaus-sian states is provided by the logarithmic negativity EN =max{0,−ln(2˜ n−)} [2]. For symmetric Gaussian states, i.e.states whose standard form is characterized by α = β, an-other computable entanglement monotone is provided by theentanglement of formation [10]. However, in this subcase thetwo measures provide the same characterization of entangle-mentandarefullyequivalent. Therefore,fromnowonwewilladopt the logarithmic negativity to quantify the entanglementof two-mode Gaussian states.We now show that a generic state in standard form can bereparametrizedin terms of the Sp(4,R) invariants  (the globalpurity) and ∆, and of the Sp(2,R) ⊕ Sp(2,R) invariants  1 and 2, where  i denotes the purity of the reduced state in modei (i = 1,2). For a generic two-mode Gaussian state σsf wethus have 1 =12a,  2 =12b, (7)116 2 = Detσ = (ab)2 − ab(c2+ + c2−) + (c+c−)2 , (8)∆ = a2 + b2 + 2c+c− . (9)Eqs. (7-9) are easily inverted to provide the followingparametrizationa =12 1, b =12 2, (10)c± =12      1 2  ∆ −( 1 −  2)24 21 22 2−14 2 ± ǫ(11)with ǫ ≡18 [( 1 +  2)2 − 4 21 22∆]2 31 32−4 1 2 2 .The global and marginal purities range from 0 to 1, con-strained by the condition  ≥  1 2 , (12)a direct consequence of Heisenberg uncertainty relations. Itimplies that no Gaussian LPTP (less pure than product) statesexist, at variance with the case of two–qubit systems [11].Eqs. (5,10, 11) determine the smallest symplectic eigenvalueof the covariance matrix σ and of its partial transpose ˜ σ2n2− = ∆ − ∆2 −14 2 , 2˜ n2− = ˜ ∆ − ˜ ∆2 −14 2 ,(13)where ˜ ∆ = −∆ + 1/2 21 + 1/2 22. This parametrizationdescribes physical states if the radicals in Eqs. (11, 13) existandIneq.(4), expressingtheHeisenbergprinciple,is satisﬁed.All these conditions can be combined and recast as upper andlower bounds on the invariant ∆12 +( 1 −  2)24 21 22≤ ∆≤ min ( 1 +  2)24 21 22−12 ,14 1 +1 2  . (14)The invariant ∆ has a direct physical interpretation: at givenglobal and marginal purities, it determines the amount of en-tanglement of the state. In fact, one has∂ ˜ n2−∂ ∆     µ1, µ2,µ=12˜ ∆ ˜ ∆2 − 14µ2− 1 > 0. (15)The smallest symplecticeigenvalueof the partiallytransposedstate is strictly monotonein ∆. Thereforethe entanglementofa generic Gaussian state σ with global purity   and marginalpurities  1,2 strictly increases with decreasing ∆. Since ∆has both lower and upper bounds, due to Ineq. (14), not onlymaximallybut also minimally entangledGaussian states exist.Thisisanimportantresultconcerningtherelationbetweenen-tanglement and purity of quantum states: the entanglement ofa Gaussian state is tightly bound by the amount of global andmarginal purities, with only one remaining degree of freedomrelated to the invariant ∆.We now aim to characterize extremal (maximally or mini-mally)entangledGaussianstates forﬁxedglobalandmarginalpurities. Let us ﬁrst consider the states saturating the lowerboundin Eq. (14), which entails maximal entanglement. Theyare Gaussian maximally entangled mixed states (GMEMS),admitting the following parametrizationa =12 1, b =12 2, c± = ±12 1 1 2−1 . (16)We nowrecallthatGaussian squeezedthermalstates are statesof the form σ = SrνSr, where Sr is the symplectic repre-sentation of the two–mode squeezing operator Exp[r(a1a2 −3a†1a†2)/2], while ν = diag{n−,n−,n+,n+}. These statesare in standard form with a = n− cosh2 r + n+ sinh2 r, b =n+ cosh2 r+n− sinh2 r, c+ = −c− = (n−+n+)sinh2r/2.In the pure case (n∓ = 1/2) they reduce to two–modesqueezed vacua. We thus ﬁnd that states of the form ofEq. (16) are non–symmetrical squeezed thermal states withtanh2r = ( 1 2 −  21 22/ )1/2/( 1 +  2). These states areseparable in the range  ≤ 1 2 1 +  2 −  1 2. (17)In such a separable region in the space of purities, no en-tanglement can occur for states of the form of Eq. (16),while, outside this region, they are GMEMS. We now con-sider the class of states that saturate the upper bound inEq. (14). They determine the class of Gaussian least entan-gled mixed states (GLEMS). Violation of Ineq. (17) impliesthat 1 + 1/ 2 /4 ≤ ( 1 + 2)2/4 21 22 −1/2 . Therefore,outside the separable region, GLEMS fulﬁll∆ =14 1 +1 2 . (18)Eq. (18) expresses saturation of Heisenberg relation (4). Wethus ﬁnd that the most semiclassical states of minimum quan-tum uncertainty are Gaussian least entangled states. GLEMSin standard form are characterized byc± =18      1 2 −4 2 + 1 +1 2 −( 1 −  2)2 21 22 2 ±18       −4 1 2 + (1 +  2) 21 22 −  2( 1 +  2)2 2 2 31 32.(19)According to the PPT criterion, GLEMS are separable onlyfor   ≤  1 2/  21 +  22 −  21 22, so that in the range 1 2 1 +  2 −  1 2<   ≤ 1 2   21 +  22 −  21 22(20)both separable and entangled states can be found. The verynarrow regiondeﬁned by Ineq. (20) is the only coexistence re-gion between entangled and separable Gaussian mixed states.Furthermore, Ineq. (14) leads to the following constraint onthe purities  ≤ 1 2 1 2 + | 1 −  2|. (21)For purities which saturate Ineq. (21), GMEMS and GLEMScoincide and we have a unique class of states dependingonly on the marginal purities  1,2. They are Gaussian max-imally entangled states for ﬁxed marginals (GMEMMS). Themaximal entanglement of a Gaussian state decreases rapidlywith increasing difference of marginal purities, in analogywith ﬁnite-dimensional systems [11]. For symmetric states0 0.2 0.4 0.6 0.8 1marginal purity Μi00.20.40.60.81globalpurityΜunphysicalentangledseparable  coexistenceFIG. 1: Summary of entanglement properties of symmetric Gaus-sian states with given global and marginal purities. In the entangledregion, the average logarithmic negativity Eq. (25) is depicted, grow-ing from gray to black. The dashed area is the coexistence region ofseparable and entangled states.( 1 =  2) Ineq. (21) reduces to the trivial bound   ≤ 1 andGMEMMS reduceto puretwo–modesqueezedstates. We cansummarize the previous results in the following scheme, clas-sifying all the two-mode Gaussian physical states accordingto their degree of global and marginal purities 1 2 ≤   ≤ 1 2 1 +  2 −  1 2⇒ separable 1 2 1 +  2 −  1 2<   ≤ 1 2   21 +  22 −  21 22⇒ coexistence 1 2   21 +  22 −  21 22<   ≤ 1 2 1 2 + | 1 −  2|⇒ entangled(22)Knowledgeof the globaland marginalpurities thus accuratelycharacterizes the entanglement of Gaussian states, providingstrong sufﬁcient conditions and analytical bounds. As we willnow show, marginaland global purities allow also an accuratequantiﬁcation of entanglement. Outside the separable region,GMEMS attain maximum logarithmic negativity ENmaxENmax( 1,2, ) = −12log −1 +  1 +  22 21 22 × 1 +  2 − ( 1 +  2)2 −4 21 22  , (23)while, in the entangled region (see Eq. (22)), GLEMS acquireminimum logarithmic negativity ENminENmin( 1,2, ) = −12log 1 21+1 22−12 2 −12−  1 21+1 22−12 2 −12 2−1 2 . (24)4FIG. 2: Upper and lower bounds on the logarithmic negativity asfunctions of the global and marginal purities of symmetric Gaussianstates. The black (gray) surface represents GMEMS (GLEMS).Knowledge of ∆ (i.e. of the full covariance matrix) wouldallow for an exact quantiﬁcation of the entanglement. How-ever, we will now show that an estimate based only onthe knowledge of the experimentally measurable global andmarginal purities turns out to be quite accurate. We can infact quantify the entanglement of Gaussian states with givenglobal and marginal purities by the “average logarithmic neg-ativity” ¯ EN deﬁned as¯ EN( 1,2, ) ≡ENmax( 1,2, ) + ENmin( 1,2, )2. (25)We can then also deﬁne the relative error δ ¯ EN on ¯ EN asδ ¯ EN( 1,2, ) ≡ENmax( 1,2, ) − ENmin( 1,2, )ENmax( 1,2, ) + ENmin( 1,2, ). (26)It is easily seen that this error decreases both with increasingglobal purity and decreasing marginal purities, i.e. with in-creasing entanglement. For ease of graphical display, let usconsider the important case of symmetric Gaussian states, forwhich the reduction  1 =  2 ≡  i occurs.1 1.5 2 2.5 3Μ￿Μi00.20.40.60.81 E   NFIG. 3: The relative error δ ¯ EN Eq. (26) on the average logarithmicnegativity as a function of the ratio µ/µi, plotted at µ = 0.5.Fig. 1 shows the classiﬁcation of the entanglement of sym-metric states depending on their global and marginal purities.Notice in particular the very narrow region of coexistence ofseparable and entangled states. In Fig. 2, ENmin( i,  ) ofEq. (24) and ENmax( i,  ) of Eq. (23) are plotted versus  iand  . In the case   = 1 the upperandlower boundscorrectlycoincide, since for pure states the entanglement is completelyquantiﬁed by the marginal purity. For mixed states this is notthe case, but, as the plot shows, knowledge of the global andmarginal purities strictly bounds the entanglement both fromabove and from below.The relative error δ ¯ EN( i,  ) given by Eq. (26) is plot-ted in Fig. 3 as a function of the ratio  / i. It decays expo-nentially, and falls below 5% for   >  i. Thus detection ofgenuinely entangled states is always assured by this method,except at most for a small set of states with very weak entan-glement(stateswithEN . 1). Moreover,theaccuracyis evengreater in the general non-symmetric case  1  =  2, becausethe maximal entanglement decreases in such an instance. Theaboveanalysis demonstratesthat the average logarithmicneg-ativity ¯ EN is a reliable estimate of the logarithmic negativityEN, improving as the entanglement increases. This allowsfor an accurate quantiﬁcation of CV entanglement by knowl-edge of the global and marginal purities. The latter quantitiesmay be in turn amenable to direct experimental determinationby exploiting the technology of quantum networks [5], evenwithout homodyning[12]. The present work thus may providea powerfuloperativecharacterizationand quantiﬁcationof theentanglement of generic Gaussian states.Financial support from INFM, INFN, and MIUR under na-tional project PRIN-COFIN 2002 is acknowledged.[1] Quantum Information Theory with Continuous Variables, S. L.Braunstein and A. K. Pati Eds. (Kluwer, Dordrecht, 2002).[2] G. Vidal and R. F. Werner, Phys. Rev. A 65, 032314 (2002);J. Eisert, Ph. D. thesis (University of Potsdam, 2001); K.˙ Zyczkowski et al., Phys. Rev. A 58, 883 (1998).[3] M.-S. Kim, J. Lee, and W. J. Munro, Phys. Rev. A 66,030301(R) (2002).[4] P.van Loockand A.Furusawa, Phys.Rev. A 67, 052315 (2003).[5] A. K. Ekert et al., Phys. Rev. Lett. 88, 217901 (2002); R. Filip,Phys. Rev. A 65, 062320 (2002).[6] R. Simon, Phys. Rev. Lett. 84, 2726 (2000).[7] L.-M. Duan, et al., Phys. Rev. Lett. 84, 2722 (2000).[8] A. S. Holevo, M. Sohma, and O. Hirota, Phys. Rev. A 59, 1820(1999); A.S. Holevoand R. F.Werner, Phys.Rev. A 63, 032312(2001).[9] A. Seraﬁni, F. Illuminati, and S. De Siena, J. Phys. B: At. Mol.Opt. Phys. 37, L21 (2004).[10] G. Giedke et al., Phys. Rev. Lett. 91, 107901 (2003).[11] G. Adesso, F. Illuminati, and S. De Siena, e-printquant-ph/0307192 (2003), and Phys. Rev. A, in press.[12] J. Fiurasek and N. J. Cerf, e-print quant-ph/0311119 (2003).

Determination of continuous variable entanglement by purity measurements

Crossref

Determination of Continuous Variable Entanglement by Purity Measurements

We classify the entanglement of two-mode Gaussian states according to their degree of total and
partial mixedness.We derive exact bounds that determine maximally and minimally entangled states
for fixed global and marginal purities. This characterization allows for an experimentally reliable
estimate of continuous variable entanglement based on measurements of purity

Determination of continuous variable entanglement by purity measurements

Abstract

Similar works

Full text

Available Versions

UCL Discovery

Crossref

Archivio della Ricerca - Università di Salerno