We present a theoretical method to determine the multipartite entanglement
between different partitions of multimode, fully or partially symmetric
Gaussian states of continuous variable systems. For such states, we determine
the exact expression of the logarithmic negativity and show that it coincides
with that of equivalent two--mode Gaussian states. Exploiting this reduction,
we demonstrate the scaling of the multipartite entanglement with the number of
modes and its reliable experimental estimate by direct measurements of the
global and local purities.Comment: 4 pages, 2 figures; to be published in Phys. Rev. Let

Alessio Serafini

Fabrizio Illuminati

Gerardo Adesso

P. van Loock

English

arXiv

Crossref

Quantification and Scaling of Multipartite Entanglement in Continuous Variable Systems

We present a theoretical method to determine the multipartite entanglement between different partitions of multimode, fully or partially symmetric Gaussian states of continuous variable systems. For such states, we determine the exact expression of the logarithmic negativity and show that it coincides with that of equivalent two-mode Gaussian states. Exploiting this reduction, we demonstrate the scaling of the multipartite entanglement with the number of modes and its reliable experimental estimate by direct measurements of the global and local purities

Adesso, G

Serafini, A

Illuminati, F

UCL Discovery

PRL 93, 220504 (2004) P H Y S I C A L R E V I E W L E T T E R S week ending26 NOVEMBER 2004Quantification and Scaling of Multipartite Entanglement in ContinuousVariable SystemsGerardo Adesso, Alessio Serafini, and Fabrizio IlluminatiDipartimento di Fisica ‘‘E. R. Caianiello,’’ Universita` di Salerno, Via S. Allende, 84081 Baronissi (SA), Italy(Received 8 June 2004; published 24 November 2004)0031-9007=We present a theoretical method to determine the multipartite entanglement between differentpartitions of multimode, fully or partially symmetric Gaussian states of continuous variable systems.For such states, we determine the exact expression of the logarithmic negativity and show that itcoincides with that of equivalent two-mode Gaussian states. Exploiting this reduction, we demonstratethe scaling of the multipartite entanglement with the number of modes and its reliable experimentalestimate by direct measurements of the global and local purities.DOI: 10.1103/PhysRevLett.93.220504 PACS numbers: 03.67.Mn, 03.65.UdThe full understanding of the structure of multipartitequantum entanglement is a major scope in quantum in-formation theory that is yet to be achieved. At the ex-perimental level, it would be crucial to devise effectivestrategies to conveniently distribute the entanglementbetween different parties, depending on the needs of theaddressed information protocol. Concerning the theory,the conditions of separability for generic bipartitions ofGaussian states of continuous variable (CV) systems havebeen derived and analyzed [1–3]. However, the quanti-fication and scaling of entanglement for arbitrary statesof multipartite systems remains in general a formidabletask [4]. In this work, we present a theoretical schemeto exactly determine the multipartite entanglement ofgeneric Gaussian symmetric states (pure or mixed) ofCV systems.We consider a CV system consisting of N canonicalbosonic modes, associated with an infinite-dimensionalHilbert space and described by the vector X^ of the fieldquadrature operators. Quantum states of paramount im-portance in CVsystems are the so-called Gaussian states,i.e., states fully characterized by first and second mo-ments of the canonical operators. When addressingphysical properties invariant under local unitary trans-formations, one can neglect first moments and com-pletely characterize Gaussian states by the 2N  2Nreal covariance matrix (CM) , whose entries are ij 1=2hfX^i; X^jgi  hX^iihX^ji. The CM  must fulfill the un-certainty relation  i 	 0, with the symplectic form  ni1! and !  ij1  ij1; i; j  1; 2: Sym-plectic operations (i.e., belonging to the group Sp2N;R fS 2 SL2N;R:STS  g) acting by congruence onCMs in phase space amount to unitary operations ondensity matrices in Hilbert space. In phase space, anyN-mode Gaussian state can be written as   STS, with  diagfn1; n1; . . . nN; nNg. The set   fnig constitutesthe symplectic spectrum of and its elements must fulfillthe conditions ni 	 1, ensuring positivity of the densitymatrix % associated to . The symplectic eigenvalues nican be computed as the eigenvalues of the matrix jij.They are determined by N symplectic invariants associ-04=93(22)=220504(4)$22.50 22050ated to the characteristic polynomial of such a matrix;two global invariants which will be useful are the deter-minant Det  Qin2i and the quantity   Pin2i , whichis the sum of the determinants of all the 2 2 subma-trices of  related to each mode.The degree of mixedness of a quantum state % ischaracterized by its purity   Tr%2. For a Gaussianstate with CM  one has simply   1= Detp . As forthe entanglement, we recall that positivity of the partiallytransposed state ~%, obtained by transposing the reducedstate of only one of the subsystems, is a necessary andsufficient condition [positive partial transpose (PPT) cri-terion] of separability for N  1-mode Gaussian statesof 1 N-mode bipartitions [5,6]. In phase space, partialtransposition amounts to a mirror reflection of one quad-rature associated to the single-mode partition. If f~nig isthe symplectic spectrum of the partially transposed CM~, then a N  1-mode Gaussian state with CM  isseparable if and only if ~ni	1 8 i. A proper measure ofCV entanglement is the logarithmic negativity EN [7],which is readily computed in terms of the symplecticspectrum ~ni of ~ as EN  Pi:~ni<1 ln~ni. Such a measurequantifies the extent to which the PPT condition ~ni 	 1 isviolated.Let us first consider the 2N  2N CM N of a fullysymmetric N-mode Gaussian state (i.e,. a state invariantunder the exchange of any two modes) N 0BBBBBB@ "    ""  " ......" . ..""    " 1CCCCCCA; (1)where  and " are 2 2 submatrices. Because of thesymmetry of such a state,  and " can be put by meansof local (single-mode) symplectic operations in the form  diagfb; bg;"  diagfe1; e2g. The symplectic spec-trum N of N has then the structure (see theAppendix)4-1  2004 The American Physical SocietyPRL 93, 220504 (2004) P H Y S I C A L R E V I E W L E T T E R S week ending26 NOVEMBER 2004N  f; . . . ; |{z}N1; N g; (2)2  b e1b e2;2N  b N  1e1b N  1e2:The N  1-degenerate eigenvalue  is independent ofN, while N can be expressed as a function of the purity  Det1=2 of the single-mode reduced state and ofthe symplectic spectrum of the two-mode block 2 ,2  f;   2 g2N  NN  22 N  12N2  N  22:(3)The global purity of the fully symmetric state isN  DetN 1=2  N1 N 1; (4)and, through Eq. (3), can be directly linked to the one-and two-mode parameters. In particular, the symplecticeigenvalues  are determined in terms of the two Sp4;Rinvariants 2 and 2 by the relation [8] 22 2 22 4=22q.Next, we consider the N  1-mode Gaussian statesconstituted by generic single-mode states with CM  andfully symmetric N-mode states with CM N of the form(1). The mode with CM  is then coupled with all other Nmodes by the same 2 2 real matrix . The CM  ofsuch N  1-mode states reads  T N;   . . .|{z}N: (5)We will now show that the properties of mixedness andentanglement of these states are determined by a suitable,limited set of global and local invariants under symplec-tic (unitary) operations. Let us introduce the purity  Det1=2 of the single-mode party, the global purity  Det1=2 of the state (5), and the globalSp2N2;R invariant    N Pin2i , wherethe nis constitute the symplectic spectrum  fn1; . . . ; nN1g of the CM , and  Det 2NDet; (6)N  NDet N  1Det": (7)We are now in the position to characterize and quantifythe bipartite entanglement between the single-mode and theN-mode blockN , the multipartite entanglementbetween all the N  1 modes, and to provide an opera-tional scheme for their experimental determination interms of measurements of the global and local purities.To proceed, we must evaluate the logarithmic negativity220504by determining the partially transposed CM ~, withrespect to the partition jN, which is obtained by flip-ping the sign of Det. Mixedness and entanglement areencoded, respectively, in the symplectic spectrum of ,and of ~. It is worth noting that, of the previouslyintroduced parameters, only  is affected by the op-eration of partial transposition  !!~~, with~   Det 2NDet    2=2: (8)The symplectic spectrum   fnig (i  1; . . . ; N  1) ofthe CM  Eq. (5) is of the form (see Appendix)  f; . . . ; |{z}N1; n; ng; (9)where  is the lowest symplectic eigenvalue of thereduced two-mode state 2 . The eigenvalues n can beevaluated observing that Eqs. (4), (7), and (9) impose theidentity     N  12  N1 N 2 whichcan be used to obtain2n2    N1 N 2  N1 N 22 4N1 2s: (10)Since partial transposition leaves the N-mode symmetricblock N unchanged, the symplectic eigenvalues of ~are again of the form ~  f~nig  f; . . . ; ; ~n; ~ng,with ~n defined as in Eq. (10), but with  replaced by~ from Eq. (8). The logarithmic negativity EjNN ,quantifying the bipartite entanglement between  andN , is determined only by those symplectic eigenvaluesof ~ which satisfy ~ni < 1. Since  	 1 (because itbelongs to the symplectic spectrum of ), the entangle-ment is determined only by the eigenvalues ~n. On theother hand, the eigenvalues n of Eq. (10) can be inter-preted as the symplectic spectrum of an equivalent two-mode state of CM eq with global purity eq and invari-ant eq given byeq  N1 ; eq    N1 N 2: (11)The corresponding ~eq associated to the partially trans-posed CM ~eq reads then ~eq  eq  2=2 2=N1 N 2. By comparison with the expression ~  2=21  2=22, holding for a generic two-modestate with local purities 1 and 2 [8], one determinesthe local purities of the equivalent two-mode state eq:eq1  ; eq2  N1 N : (12)The two global invariants [Eq. (11)] and the two localinvariants [Eq. (12)] determine uniquely the entangle-ment of the two-mode Gaussian state with CM eq. Inparticular, one can immediately see that the symplecticeigenvalues of the partially transposed CM ~eq coincidewith ~n, so that we obtain the crucial result that the-21 2 3 4 5 6 7 8b ≡ 1/µΒ ∼ squeezing00.511.522.531×KtnemelgnatneE Nβ|βKK=1K=3K=5K=7K=8K=9FIG. 1 (color online). Entanglement hierarchy for N1-mode GHZ-type states (N  9).PRL 93, 220504 (2004) P H Y S I C A L R E V I E W L E T T E R S week ending26 NOVEMBER 2004logarithmic negativity of the equivalent two-mode statecoincides with the logarithmic negativity EjNN of theN  1-mode state. Explicitly, one hasEjNN  maxf0; log~ng; (13)with 2~n2  ~eq ~2eq  4=2eqq. Indeed, only thesmallest symplectic eigenvalue ~n enters in the determi-nation of the multimode entanglement, since ~n > 1 fortwo-mode states [8].The 1 N entanglement is completely quantified bymeasuring the two local purities  and N , the globalpurity , the symplectic eigenvalue , and Det(which together with  determines ). For two-mode Gaussian states, a reliable quantitative estimate ofthe logarithmic negativity, yielding exact (and very nar-row) lower and upper bounds on the entanglement, can beobtained by simply measuring the global and local puri-ties of the state [8]. In the present instance, this factimplies that a reliable estimate of the 1 N entanglementdoes not require the knowledge of the correlation matrix, while the remaining four quantities (the three puritiesand the eigenvalue ) can be measured even withouthomodyning by direct single-photon detections [9].Moreover, knowledge of these few quantities is also suf-ficient to determine the multimode, multipartite entan-glement of the state . In fact, the fully symmetricN-mode block N can be again regarded as a statedescribing a mode with CM  coupled with a fullysymmetric N  1-mode block N1 , and thus the 1N  1 entanglement within N can again be computedby constructing the corresponding equivalent two-modestate and evaluating its entanglement. This scaling pro-cedure can be iterated to determine all the multimodeentanglements existing between each mode and each fullysymmetric K-mode sub-block K (K  1; . . . ; N). The1 K entanglement between the single-mode  and anyfully symmetric K-mode partition K of N can bedetermined in a similar way. A crucial feature of thisscaling structure of the multipartite entanglement is that,at every step of the cascade, the 1 K entanglement isalways equivalent to a 1 1 entanglement, so that thequantum correlations between the different partitions of can be directly compared to each other; it is thuspossible to establish a multimode entanglement hierarchywithout any problem of ordering.To illustrate the scaling structure of multipartite en-tanglement in CV systems let us consider a pure, N 1-mode fully symmetric Gaussian state of the form ofEq. (1). Imposing the constraint of pure state (  1,  N1  1), one obtains ei  f1 b2N 1N1i b2 1b2N 12N 12p g=2bN. Sucha state belongs to the class of CV Greenberger-Horne-Zeilinger (GHZ)-type states discussed in Ref. [3]. Thesemultipartite entangled states are the outputs of a sequence22050of N beam splitters with N  1 single-mode squeezedinputs [3,10]. In the limit of infinite squeezing, thesestates reduce to the proper GHZ states of CV systems[3]. The CM GHZN1 of this class of pure states, for a givennumber of modes, depends only on the parameter b 1= 	 1, which is an increasing function of the single-mode squeezing. Correlations between the modes areinduced according to the above expression for the cova-riances ei. Exploiting our previous analysis, we can com-pute the entanglement between a single-mode withreduced CM  and any K-mode partition of the remain-ing modes (1  K  N), by determining the equivalenttwo-mode CMeqjK . The 1 K entanglement quantifiedby the logarithmic negativity EjKN is determined by thesmallest symplectic eigenvalue ~nK;N of the partiallytransposed CM ~eqjK . For any nonzero squeezing (i.e.,b > 1) one has that ~nK;N < 1, meaning that the state ex-hibits genuine multipartite entanglement; each mode isentangled with any other K-mode block, as first remarkedin Ref. [3]. Further, the genuine multipartite nature of theentanglement can be precisely quantified by observingthat EjKN 	EjK1N , as shown in Fig. 1. The 1 1 en-tanglement between two modes is weaker than the 1 2one between a mode and other two modes, which is inturn weaker than the 1 K one, and so on with increas-ing K in this typical cascade structure. From an opera-tional point of view, the signature of genuine multipartiteentanglement is revealed by the fact that performing, e.g.,a local measurement on a single-mode will affect all theother N modes. This means that the quantum correlationscontained in the state with CM GHZN1 can be fully recov-ered only when considering the 1 N partition. In par-ticular, the pure-state 1 N logarithmic negativity is, asexpected, independent of N, being a simple monotonicfunction of the entropy of entanglement EV (defined as thevon Neumann entropy of the reduced single-mode statewith CM ). It is worth noting that, in the limit of infinitesqueezing (b! 1), only the 1 N entanglement di-verges while all the other 1 K quantum correlationsremain finite (see Fig. 1). Namely, EjKN GHZN1 !b!11=2 logf1 4K=NK  1  KK  3g, which can-4-32 3 4 510 502001000infinity00.10.20.30.40.50.60.70.80.91Logarithmic Negativity # of modes (N+1)1x11x(N–1)FIG. 2. Scaling as a function of N of the 1 1 and of the1 N  1 entanglement for a N  1-mode GHZ-type CVpure-state (b  1:5).PRL 93, 220504 (2004) P H Y S I C A L R E V I E W L E T T E R S week ending26 NOVEMBER 2004not exceed log5p ’ 0:8 for any N and for any K <N. Atfixed squeezing, the scaling with N of the 1 N  1entanglement compared to the 1 1 entanglement isshown in Fig. 2 (we recall that the 1 N entanglementis independent on N). Notice how, with increasing num-ber of modes, the multipartite entanglement increases tothe detriment of the two-mode one which becomes dis-tributed between all the modes. We remark that this scal-ing occurs in any Gaussian states, either fully or partiallysymmetric, pure or mixed. For instance, this is the casefor a single-mode squeezed state coupled with a N-modesymmetric thermal squeezed state. The simplest exampleof a mixed state with genuine multipartite entanglementis obtained from GHZN1 by tracing out some of the modes.Figure 2 can then also be seen as a demonstration of thescaling in such a N-mode mixed state, where the 1N  1 entanglement is the strongest one. Thus, withincreasing N, the global mixedness can limit but notdestroy the genuine multipartite entanglement betweenall the modes. This entanglement is experimentally ac-cessible by all-optical means [3], and it also allows for areliable (i.e., with fidelity F > 1=2) quantum teleporta-tion between any two parties [10]. Therefore, the quanti-fication of multipartite entanglement by measurements ofpurity, which, as we have already remarked, can beexperimentally implemented even without homodyning,leads to an accurate estimate of the multiparty teleporta-tion efficiency and to direct control on the transfer ofquantum information.In conclusion, we have shown that multipartite quan-tum correlations of Gaussian states of 1 N bipartitionsunder symmetry are endowed with a scaling structurethat reduces the problem to the analysis of the entangle-ment of equivalent two-mode Gaussian states. Thanks tothis reduction, it is possible to determine exactly thelogarithmic negativity of the multimode states and toallow for a reliable experimental estimate of the multi-partite entanglement by direct measurements of globaland local purities, without the need for the full recon-220504truction of the covariance matrix. Our results apply tomany cases of practical interest. For instance, the entireclass of bi-symmetric—i.e., invariant under the ex-change of two given modes— three-mode Gaussian states[2] has its multipartite entanglement completely quanti-fied by the present analysis. The generalization of thepresent approach for the quantification of multipartiteCV entanglement to states with weaker symmetry con-straints and to M N-mode bipartitions (with M> 1)awaits further study.Financial support from INFM, INFN, and MIUR isacknowledged.Appendix: Proof of the symplectic degeneracy.—Weprove here the multiplicity of the symplectic eigenvalue for the CMs N and , asserted in Eqs. (2) and (9) .We first recall that, if   f1; . . . ; Ng is the symplecticspectrum of the CM , then the 2N eigenvalues of thematrix i are given by the set fig. Let us focus nexton the CM 2 : in the linear space on which the matrixi2 acts, the eigenvector v corresponding to the ei-genvalue  reads vibe1 ;1;ibe1;1T. Becauseof the symmetry of N , any 2N-dimensional vector v ofthe formv  0; . . . 0;i b e1;1|{z}mode i; 0 . . . 0; ib e1; 1|{z}mode j; 0; . . . 0T(1a)(i.e., any vector obtained by taking v in a couple ofmodes ij and appending to it 0 elements for all the othermodes) is an eigenvector of iN with eigenvalue . Itcan be seen immediately that one can construct N  1linear independent vectors of the above form, provingEq. (2). Clearly, an analogous reasoning holds for thematrix , proving Eq. (9).-4[1] G. Giedke et al., Phys. Rev. Lett. 87, 167904 (2001).[2] G. Giedke et al., Phys. Rev. A 64, 052303 (2001).[3] P. van Loock and A. Furusawa, Phys. Rev. A 67, 052315(2003).[4] P. van Loock and S. L. Braunstein, in QuantumInformation Theory with Continuous Variables, editedby S. L. Braunstein and A. K. Pati (Kluwer, Dordrecht,2002).[5] R. Simon, Phys. Rev. Lett. 84, 2726 (2000).[6] R. F. Werner and M. M. Wolf, Phys. Rev. Lett. 86, 3658(2001).[7] G.Vidal and R. F. Werner, Phys. Rev. A 65, 032314 (2002).[8] G. Adesso, A. Serafini, and F. Illuminati, Phys. Rev. Lett.92, 087901 (2004); Phys. Rev. A 70, 022318 (2004).[9] J. Fiura´sˇek and N. J. Cerf, Phys. Rev. Lett. 93, 063601(2004); J. Wenger et al., Phys. Rev. A 70, 053812 (2004).[10] P. van Loock and S. L. Braunstein, Phys. Rev. Lett. 84,3482 (2000).

Quantification and scaling of multipartite entanglement in continuous variable systems

ADESSO, GERARDO

ILLUMINATI, Fabrizio

A. SERAFINI

Archivio della Ricerca - Università di Salerno

PRL 93, 220504 (2004) P H Y S I C A L R E V I E W L E T T E R Sweek ending26 NOVEMBER 2004Quantification and Scaling of Multipartite Entanglement in ContinuousVariable SystemsGerardo Adesso, Alessio Serafini, and Fabrizio IlluminatiDipartimento di Fisica ‘‘E. R. Caianiello,’’ Università di Salerno, Via S. Allende, 84081 Baronissi (SA), Italy(Received 8 June 2004; published 24 November 2004)0031-9007=We present a theoretical method to determine the multipartite entanglement between differentpartitions of multimode, fully or partially symmetric Gaussian states of continuous variable systems.For such states, we determine the exact expression of the logarithmic negativity and show that itcoincides with that of equivalent two-mode Gaussian states. Exploiting this reduction, we demonstratethe scaling of the multipartite entanglement with the number of modes and its reliable experimentalestimate by direct measurements of the global and local purities.DOI: 10.1103/PhysRevLett.93.220504 PACS numbers: 03.67.Mn, 03.65.UdThe full understanding of the structure of multipartitequantum entanglement is a major scope in quantum in-formation theory that is yet to be achieved. At the ex-perimental level, it would be crucial to devise effectivestrategies to conveniently distribute the entanglementbetween different parties, depending on the needs of theaddressed information protocol. Concerning the theory,the conditions of separability for generic bipartitions ofGaussian states of continuous variable (CV) systems havebeen derived and analyzed [1–3]. However, the quanti-fication and scaling of entanglement for arbitrary statesof multipartite systems remains in general a formidabletask [4]. In this work, we present a theoretical schemeto exactly determine the multipartite entanglement ofgeneric Gaussian symmetric states (pure or mixed) ofCV systems.We consider a CV system consisting of N canonicalbosonic modes, associated with an infinite-dimensionalHilbert space and described by the vector X̂ of the fieldquadrature operators. Quantum states of paramount im-portance in CVsystems are the so-called Gaussian states,i.e., states fully characterized by first and second mo-ments of the canonical operators. When addressingphysical properties invariant under local unitary trans-formations, one can neglect first moments and com-pletely characterize Gaussian states by the 2N  2Nreal covariance matrix (CM) , whose entries are ij 1=2hfX̂i; X̂jgi  hX̂iihX̂ji. The CM  must fulfill the un-certainty relation  i 	 0, with the symplectic form  ni1! and !  ij1  ij1; i; j  1; 2: Sym-plectic operations (i.e., belonging to the group Sp2N;R fS 2 SL2N;R:STS  g) acting by congruence onCMs in phase space amount to unitary operations ondensity matrices in Hilbert space. In phase space, anyN-mode Gaussian state can be written as   STS, with  diagfn1; n1; . . . nN; nNg. The set   fnig constitutesthe symplectic spectrum of  and its elements must fulfillthe conditions ni 	 1, ensuring positivity of the densitymatrix % associated to . The symplectic eigenvalues nican be computed as the eigenvalues of the matrix jij.They are determined by N symplectic invariants associ-04=93(22)=220504(4)$22.50 22050ated to the characteristic polynomial of such a matrix;two global invariants which will be useful are the deter-minant Det Qin2i and the quantity  Pin2i , whichis the sum of the determinants of all the 2 2 subma-trices of  related to each mode.The degree of mixedness of a quantum state % ischaracterized by its purity   Tr%2. For a Gaussianstate with CM  one has simply   1=Detp. As forthe entanglement, we recall that positivity of the partiallytransposed state ~%, obtained by transposing the reducedstate of only one of the subsystems, is a necessary andsufficient condition [positive partial transpose (PPT) cri-terion] of separability for N  1-mode Gaussian statesof 1 N-mode bipartitions [5,6]. In phase space, partialtransposition amounts to a mirror reflection of one quad-rature associated to the single-mode partition. If f~nig isthe symplectic spectrum of the partially transposed CM~, then a N  1-mode Gaussian state with CM  isseparable if and only if ~ni	1 8 i. A proper measure ofCV entanglement is the logarithmic negativity EN [7],which is readily computed in terms of the symplecticspectrum ~ni of ~ as EN  Pi:~ni<1 ln~ni. Such a measurequantifies the extent to which the PPT condition ~ni 	 1 isviolated.Let us first consider the 2N  2N CM N of a fullysymmetric N-mode Gaussian state (i.e,. a state invariantunder the exchange of any two modes) N 0BBBBBB@ "    ""  " ......" . ..""    " 1CCCCCCA; (1)where  and " are 2 2 submatrices. Because of thesymmetry of such a state,  and " can be put by meansof local (single-mode) symplectic operations in the form  diagfb; bg;"  diagfe1; e2g. The symplectic spec-trum N of N has then the structure (see theAppendix)4-1  2004 The American Physical SocietyPRL 93, 220504 (2004) P H Y S I C A L R E V I E W L E T T E R Sweek ending26 NOVEMBER 2004N  f; . . . ; |{z}N1; N g; (2)2  b e1b e2;2N b N  1e1b N  1e2:The N  1-degenerate eigenvalue  is independent ofN, while N can be expressed as a function of the purity  Det1=2 of the single-mode reduced state and ofthe symplectic spectrum of the two-mode block 2 ,2  f;   2 g2N NN  22N  12N2  N  22:(3)The global purity of the fully symmetric state isN  DetN 1=2  N1 N 1; (4)and, through Eq. (3), can be directly linked to the one-and two-mode parameters. In particular, the symplecticeigenvalues  are determined in terms of the two Sp4;Rinvariants 2 and 2 by the relation [8] 22 2 22 4=22q.Next, we consider the N  1-mode Gaussian statesconstituted by generic single-mode states with CM  andfully symmetric N-mode states with CM N of the form(1). The mode with CM  is then coupled with all other Nmodes by the same 2 2 real matrix . The CM  ofsuch N  1-mode states reads  T N;   . . .|{z}N: (5)We will now show that the properties of mixedness andentanglement of these states are determined by a suitable,limited set of global and local invariants under symplec-tic (unitary) operations. Let us introduce the purity  Det1=2 of the single-mode party, the global purity  Det1=2 of the state (5), and the globalSp2N2;R invariant    N Pin2i , wherethe nis constitute the symplectic spectrum  fn1; . . . ; nN1g of the CM , and  Det 2NDet; (6)N  NDet N  1Det": (7)We are now in the position to characterize and quantifythe bipartite entanglement between the single-mode and the N-mode block N , the multipartite entanglementbetween all the N  1 modes, and to provide an opera-tional scheme for their experimental determination interms of measurements of the global and local purities.To proceed, we must evaluate the logarithmic negativity220504by determining the partially transposed CM ~, withrespect to the partition jN, which is obtained by flip-ping the sign of Det. Mixedness and entanglement areencoded, respectively, in the symplectic spectrum of ,and of ~. It is worth noting that, of the previouslyintroduced parameters, only  is affected by the op-eration of partial transposition  !!~~, with~   Det 2NDet    2=2: (8)The symplectic spectrum   fnig (i  1; . . . ; N  1) ofthe CM  Eq. (5) is of the form (see Appendix)  f; . . . ; |{z}N1; n; ng; (9)where  is the lowest symplectic eigenvalue of thereduced two-mode state 2 . The eigenvalues n can beevaluated observing that Eqs. (4), (7), and (9) impose theidentity     N  12  N1 N 2 whichcan be used to obtain2n2    N1 N 2  N1 N 22 4N1 2s: (10)Since partial transposition leaves the N-mode symmetricblock N unchanged, the symplectic eigenvalues of ~are again of the form ~  f~nig  f; . . . ; ; ~n; ~ng,with ~n defined as in Eq. (10), but with  replaced by~ from Eq. (8). The logarithmic negativity EjNN ,quantifying the bipartite entanglement between  andN , is determined only by those symplectic eigenvaluesof ~ which satisfy ~ni < 1. Since  	 1 (because itbelongs to the symplectic spectrum of ), the entangle-ment is determined only by the eigenvalues ~n. On theother hand, the eigenvalues n of Eq. (10) can be inter-preted as the symplectic spectrum of an equivalent two-mode state of CM eq with global purity eq and invari-ant eq given byeq  N1 ; eq    N1 N 2: (11)The corresponding ~eq associated to the partially trans-posed CM ~eq reads then ~eq  eq  2=2 2=N1 N 2. By comparison with the expression ~  2=21  2=22, holding for a generic two-modestate with local purities 1 and 2 [8], one determinesthe local purities of the equivalent two-mode state eq:eq1  ; eq2  N1 N : (12)The two global invariants [Eq. (11)] and the two localinvariants [Eq. (12)] determine uniquely the entangle-ment of the two-mode Gaussian state with CM eq. Inparticular, one can immediately see that the symplecticeigenvalues of the partially transposed CM ~eq coincidewith ~n, so that we obtain the crucial result that the-21 2 3 4 5 6 7 8b ≡ 1/µΒ ∼ squeezing00.511.522.531×KtnemelgnatneENβ|βKK=1K=3K=5K=7K=8K=9FIG. 1 (color online). Entanglement hierarchy for N1-mode GHZ-type states (N  9).PRL 93, 220504 (2004) P H Y S I C A L R E V I E W L E T T E R Sweek ending26 NOVEMBER 2004logarithmic negativity of the equivalent two-mode statecoincides with the logarithmic negativity EjNN of theN  1-mode state. Explicitly, one hasEjNN  maxf0; log~ng; (13)with 2~n2  ~eq ~2eq  4=2eqq. Indeed, only thesmallest symplectic eigenvalue ~n enters in the determi-nation of the multimode entanglement, since ~n > 1 fortwo-mode states [8].The 1 N entanglement is completely quantified bymeasuring the two local purities  and N , the globalpurity , the symplectic eigenvalue , and Det(which together with  determines ). For two-mode Gaussian states, a reliable quantitative estimate ofthe logarithmic negativity, yielding exact (and very nar-row) lower and upper bounds on the entanglement, can beobtained by simply measuring the global and local puri-ties of the state [8]. In the present instance, this factimplies that a reliable estimate of the 1 N entanglementdoes not require the knowledge of the correlation matrix, while the remaining four quantities (the three puritiesand the eigenvalue ) can be measured even withouthomodyning by direct single-photon detections [9].Moreover, knowledge of these few quantities is also suf-ficient to determine the multimode, multipartite entan-glement of the state . In fact, the fully symmetricN-mode block N can be again regarded as a statedescribing a mode with CM  coupled with a fullysymmetric N  1-mode block N1 , and thus the 1N  1 entanglement within N can again be computedby constructing the corresponding equivalent two-modestate and evaluating its entanglement. This scaling pro-cedure can be iterated to determine all the multimodeentanglements existing between each mode and each fullysymmetric K-mode sub-block K (K  1; . . . ; N). The1 K entanglement between the single-mode  and anyfully symmetric K-mode partition K of N can bedetermined in a similar way. A crucial feature of thisscaling structure of the multipartite entanglement is that,at every step of the cascade, the 1 K entanglement isalways equivalent to a 1 1 entanglement, so that thequantum correlations between the different partitions of can be directly compared to each other; it is thuspossible to establish a multimode entanglement hierarchywithout any problem of ordering.To illustrate the scaling structure of multipartite en-tanglement in CV systems let us consider a pure, N 1-mode fully symmetric Gaussian state of the form ofEq. (1). Imposing the constraint of pure state (  1 ,  N1  1), one obtains ei  f1 b2N 1N1ib2  1b2N 12 N 12pg=2bN. Sucha state belongs to the class of CV Greenberger-Horne-Zeilinger (GHZ)-type states discussed in Ref. [3]. Thesemultipartite entangled states are the outputs of a sequence22050of N beam splitters with N  1 single-mode squeezedinputs [3,10]. In the limit of infinite squeezing, thesestates reduce to the proper GHZ states of CV systems[3]. The CM GHZN1of this class of pure states, for a givennumber of modes, depends only on the parameter b 1= 	 1, which is an increasing function of the single-mode squeezing. Correlations between the modes areinduced according to the above expression for the cova-riances ei. Exploiting our previous analysis, we can com-pute the entanglement between a single-mode withreduced CM  and any K-mode partition of the remain-ing modes (1  K  N), by determining the equivalenttwo-mode CM eqjK . The 1 K entanglement quantifiedby the logarithmic negativity EjKN is determined by thesmallest symplectic eigenvalue ~nK;N of the partiallytransposed CM ~eqjK . For any nonzero squeezing (i.e.,b > 1) one has that ~nK;N < 1, meaning that the state ex-hibits genuine multipartite entanglement; each mode isentangled with any other K-mode block, as first remarkedin Ref. [3]. Further, the genuine multipartite nature of theentanglement can be precisely quantified by observingthat EjKN 	EjK1N , as shown in Fig. 1. The 1 1 en-tanglement between two modes is weaker than the 1 2one between a mode and other two modes, which is inturn weaker than the 1 K one, and so on with increas-ing K in this typical cascade structure. From an opera-tional point of view, the signature of genuine multipartiteentanglement is revealed by the fact that performing, e.g.,a local measurement on a single-mode will affect all theother N modes. This means that the quantum correlationscontained in the state with CM GHZN1can be fully recov-ered only when considering the 1 N partition. In par-ticular, the pure-state 1 N logarithmic negativity is, asexpected, independent of N, being a simple monotonicfunction of the entropy of entanglement EV (defined as thevon Neumann entropy of the reduced single-mode statewith CM ). It is worth noting that, in the limit of infinitesqueezing (b ! 1), only the 1 N entanglement di-verges while all the other 1 K quantum correlationsremain finite (see Fig. 1). Namely, EjKN GHZN1 !b!11=2 logf1 4K=NK  1  KK  3g, which can-4-32 3 4 510 502001000infinity00.10.20.30.40.50.60.70.80.91Logarithmic Negativity # of modes (N+1)1x11x(N–1)FIG. 2. Scaling as a function of N of the 1 1 and of the1 N  1 entanglement for a N  1-mode GHZ-type CVpure-state (b  1:5).PRL 93, 220504 (2004) P H Y S I C A L R E V I E W L E T T E R Sweek ending26 NOVEMBER 2004not exceed log5p’ 0:8 for any N and for any K <N. Atfixed squeezing, the scaling with N of the 1 N  1entanglement compared to the 1 1 entanglement isshown in Fig. 2 (we recall that the 1 N entanglementis independent on N). Notice how, with increasing num-ber of modes, the multipartite entanglement increases tothe detriment of the two-mode one which becomes dis-tributed between all the modes. We remark that this scal-ing occurs in any Gaussian states, either fully or partiallysymmetric, pure or mixed. For instance, this is the casefor a single-mode squeezed state coupled with a N-modesymmetric thermal squeezed state. The simplest exampleof a mixed state with genuine multipartite entanglementis obtained from GHZN1by tracing out some of the modes.Figure 2 can then also be seen as a demonstration of thescaling in such a N-mode mixed state, where the 1N  1 entanglement is the strongest one. Thus, withincreasing N, the global mixedness can limit but notdestroy the genuine multipartite entanglement betweenall the modes. This entanglement is experimentally ac-cessible by all-optical means [3], and it also allows for areliable (i.e., with fidelity F > 1=2) quantum teleporta-tion between any two parties [10]. Therefore, the quanti-fication of multipartite entanglement by measurements ofpurity, which, as we have already remarked, can beexperimentally implemented even without homodyning,leads to an accurate estimate of the multiparty teleporta-tion efficiency and to direct control on the transfer ofquantum information.In conclusion, we have shown that multipartite quan-tum correlations of Gaussian states of 1 N bipartitionsunder symmetry are endowed with a scaling structurethat reduces the problem to the analysis of the entangle-ment of equivalent two-mode Gaussian states. Thanks tothis reduction, it is possible to determine exactly thelogarithmic negativity of the multimode states and toallow for a reliable experimental estimate of the multi-partite entanglement by direct measurements of globaland local purities, without the need for the full recon-220504truction of the covariance matrix. Our results apply tomany cases of practical interest. For instance, the entireclass of bi-symmetric—i.e., invariant under the ex-change of two given modes— three-mode Gaussian states[2] has its multipartite entanglement completely quanti-fied by the present analysis. The generalization of thepresent approach for the quantification of multipartiteCV entanglement to states with weaker symmetry con-straints and to M N-mode bipartitions (with M> 1)awaits further study.Financial support from INFM, INFN, and MIUR isacknowledged.Appendix: Proof of the symplectic degeneracy.—Weprove here the multiplicity of the symplectic eigenvalue for the CMs N and , asserted in Eqs. (2) and (9) .We first recall that, if   f1; . . . ; Ng is the symplecticspectrum of the CM , then the 2N eigenvalues of thematrix i are given by the set fig. Let us focus nexton the CM 2 : in the linear space on which the matrixi2 acts, the eigenvector v corresponding to the ei-genvalue  reads vibe1;1;ibe1 ;1T. Becauseof the symmetry of N , any 2N-dimensional vector v ofthe formv  0; . . . 0;ib e1;1|{z}mode i; 0 . . . 0; ib e1; 1|{z}mode j; 0; . . . 0T(1a)(i.e., any vector obtained by taking v in a couple ofmodes ij and appending to it 0 elements for all the othermodes) is an eigenvector of iN with eigenvalue . Itcan be seen immediately that one can construct N  1linear independent vectors of the above form, provingEq. (2). Clearly, an analogous reasoning holds for thematrix , proving Eq. (9).-4[1] G. Giedke et al., Phys. Rev. Lett. 87, 167904 (2001).[2] G. Giedke et al., Phys. Rev. A 64, 052303 (2001).[3] P. van Loock and A. Furusawa, Phys. Rev. A 67, 052315(2003).[4] P. van Loock and S. L. Braunstein, in QuantumInformation Theory with Continuous Variables, editedby S. L. Braunstein and A. K. Pati (Kluwer, Dordrecht,2002).[5] R. Simon, Phys. Rev. Lett. 84, 2726 (2000).[6] R. F. Werner and M. M. Wolf, Phys. Rev. Lett. 86, 3658(2001).[7] G.Vidal and R. F. Werner, Phys. Rev. A 65, 032314 (2002).[8] G. Adesso, A. Serafini, and F. Illuminati, Phys. Rev. Lett.92, 087901 (2004); Phys. Rev. A 70, 022318 (2004).[9] J. Fiurášek and N. J. Cerf, Phys. Rev. Lett. 93, 063601(2004); J. Wenger et al., Phys. Rev. A 70, 053812 (2004).[10] P. van Loock and S. L. Braunstein, Phys. Rev. Lett. 84,3482 (2000).

Quantification and scaling of multipartite entanglement in continuous
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Quantification and scaling of multipartite entanglement in continuous variable systems

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