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Bipartite and multipartite entanglement of Gaussian states

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Abstract: In this chapter we review the characterization of entanglement in Gaussian states of continuous variable systems. For two-mode Gaussian states, we discuss how their bipartite entanglement can be accurately quantified in terms of the global
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    a  r   X   i  v  :  q  u  a  n   t  -  p   h   /   0   5   1   0   0   5   2  v   2   6   M  a  r   2   0   0   7 February 1, 2008 10:33 WSPC/Trim Size: 9in x 6in for Review Volume AdessoIlluminati˙NEW CHAPTER 1Bipartite and Multipartite Entanglement of Gaussian States Gerardo Adesso and Fabrizio Illuminati Dipartimento di Fisica “E. R. Caianiello”, Universit`a di Salerno;CNR-Coherentia, Gruppo di Salerno; and INFN Sezione di Napoli-GruppoCollegato di Salerno, Via S. Allende, 84081 Baronissi (SA), Italy E-mail: gerardo@sa.infn.it, illuminati@sa.infn.it  In this chapter we review the characterization of entanglement in Gaus-sian states of continuous variable systems. For two-mode Gaussian states,we discuss how their bipartite entanglement can be accurately quan-tified in terms of the global and local amounts of mixedness, and ef-ficiently estimated by direct measurements of the associated purities.For multimode Gaussian states endowed with local symmetry with re-spect to a given bipartition, we show how the multimode block entan-glement can be completely and reversibly localized onto a single pairof modes by local, unitary operations. We then analyze the distributionof entanglement among multiple parties in multimode Gaussian states.We introduce the continuous-variable tangle to quantify entanglementsharing in Gaussian states and we prove that it satisfies the Coffman-Kundu-Wootters monogamy inequality. Nevertheless, we show that pure,symmetric three–mode Gaussian states, at variance with their discrete-variable counterparts, allow a promiscuous sharing of quantum corre-lations, exhibiting both maximum tripartite residual entanglement andmaximum couplewise entanglement between any pair of modes. Finally,we investigate the connection between multipartite entanglement and theoptimal fidelity in a continuous-variable quantum teleportation network.We show how the fidelity can be maximized in terms of the best prepara-tion of the shared entangled resources and, viceversa, that this optimalfidelity provides a clearcut operational interpretation of several measuresof bipartite and multipartite entanglement, including the entanglementof formation, the localizable entanglement, and the continuous-variabletangle. 1  February 1, 2008 10:33 WSPC/Trim Size: 9in x 6in for Review Volume AdessoIlluminati˙NEW2  G. Adesso and F. Illuminati  1. Introduction One of the main challenges in fundamental quantum theory as well asin quantum information and computation sciences lies in the characteriza-tion and quantification of bipartite entanglement for mixed states, and inthe definition and interpretation of multipartite entanglement both for purestates and in the presence of mixedness. While important insights have beengained on these issues in the context of qubit systems, a less satisfactoryunderstanding has been achieved until recent times on higher-dimensionalsystems, as the structure of entangled states in Hilbert spaces of high di-mensionality exhibits a formidable degree of complexity. However, and quiteremarkably, in infinite-dimensional Hilbert spaces of continuous-variablesystems, ongoing and coordinated efforts by different research groups haveled to important progresses in the understanding of the entanglement prop-erties of a restricted class of states, the so-called Gaussian states. Thesestates, besides being of great importance both from a fundamental pointof view and in practical applications, share peculiar features that maketheir structural properties amenable to accurate and detailed theoreticalanalysis. It is the aim of this chapter to review some of the most recentresults on the characterization and quantification of bipartite and multi-partite entanglement in Gaussian states of continuous variable systems,their relationships with standard measures of purity and mixedness, andtheir operational interpretations in practical applications such as quantumcommunication, information transfer, and quantum teleportation. 2. Gaussian States of Continuous Variable Systems We consider a continuous variable (CV) system consisting of   N   canon-ical bosonic modes, associated to an infinite-dimensional Hilbert space  H and described by the vector ˆ X   =  { ˆ x 1 ,  ˆ  p 1 ,...,  ˆ x N  ,  ˆ  p N  }  of the field quadra-ture (“position” and “momentum”) operators. The quadrature phase op-erators are connected to the annihilation ˆ a i  and creation ˆ a † i  operators of each mode, by the relations ˆ x i  = (ˆ a i  + ˆ a † i ) and ˆ  p i  = (ˆ a i  −  ˆ a † i ) /i . Thecanonical commutation relations for the ˆ X  i ’s can be expressed in ma-trix form: [ ˆ X  i ,  ˆ X  j ] = 2 i Ω ij , with the symplectic form Ω =  ⊕ ni =1 ω  and ω  =  δ  ij − 1 − δ  ij +1 , i,j  = 1 , 2.Quantum states of paramount importance in CV systems are the so-called Gaussian states,  i.e.  states with Gaussian characteristic functionsand quasi–probability distributions1. The interest in this special class of states (important examples include vacua, coherent, squeezed, thermal, andsqueezed-thermal states of the electromagnetic field) stems from the feasi-  February 1, 2008 10:33 WSPC/Trim Size: 9in x 6in for Review Volume AdessoIlluminati˙NEW Bipartite and Multipartite Entanglement of Gaussian States  3 bility to produce and control them with linear optical elements, and fromthe increasing number of efficient proposals and successful experimental im-plementations of CV quantum information and communication processesinvolving multimode Gaussian states (see Ref. 2 for recent reviews). Bydefinition, a Gaussian state is completely characterized by first and sec-ond moments of the canonical operators. When addressing physical prop-erties invariant under local unitary transformations, such as mixedness andentanglement, one can neglect first moments and completely characterizeGaussian states by the 2 N  × 2 N   real covariance matrix (CM)  σ , whose en-tries are  σ ij  = 1 / 2 { ˆ X  i ,  ˆ X  j }− ˆ X  i  ˆ X  j  . Throughout this chapter,  σ  willbe used indifferently to indicate the CM of a Gaussian state or the stateitself. A real, symmetric matrix  σ  must fulfill the Robertson-Schr¨odingeruncertainty relation3 σ  + i Ω ≥ 0 ,  (1)to be a  bona fide   CM of a physical state. Symplectic operations ( i.e.  be-longing to the group  Sp (2 N, R )  =  { S   ∈  SL (2 N, R ) :  S  T  Ω S   = Ω } ) actingby congruence on CMs in phase space, amount to unitary operations ondensity matrices in Hilbert space. In phase space, any  N  -mode Gaussianstate can be transformed by symplectic operations in its Williamson di-agonal form4 ν  , such that  σ  =  S  T  ν  S  , with  ν   = diag { ν  1 ,ν  1 ,...ν  N  ,ν  N  } .The set Σ =  { ν  i }  of the positive-defined eigenvalues of   | i Ω σ |  constitutesthe symplectic spectrum of   σ  and its elements, the so-called symplecticeigenvalues, must fulfill the conditions  ν  i  ≥  1, following from Eq. (1) andensuring positivity of the density matrix associated to  σ . We remark thatthe full saturation of the uncertainty principle can only be achieved bypure  N  -mode Gaussian states, for which  ν  i  = 1  ∀ i  = 1 ,...,N  . Instead,those mixed states such that  ν  i ≤ k  = 1 and  ν  i>k  >  1, with 1  ≤  k  ≤  N  ,partially saturate the uncertainty principle, with partial saturation becom-ing weaker with decreasing  k . The symplectic eigenvalues  ν  i  are determinedby  N   symplectic invariants associated to the characteristic polynomial of the matrix | i Ω σ | . Global invariants include the determinant Det σ  =  i ν  2 i and the quantity ∆ =   i ν  2 i  , which is the sum of the determinants of allthe 2 × 2 submatrices of   σ  related to each mode5.The degree of information about the preparation of a quantum state  ̺ can be characterized by its  purity   µ  ≡  Tr ̺ 2 , ranging from 0 (completelymixed states) to 1 (pure states). For a Gaussian state with CM  σ  one has6 µ  = 1 / √  Det σ .  (2)As for the entanglement, we recall that positivity of the CM’s partialtranspose (PPT)7is a necessary and sufficient condition of separability  February 1, 2008 10:33 WSPC/Trim Size: 9in x 6in for Review Volume AdessoIlluminati˙NEW4  G. Adesso and F. Illuminati  for ( M   +  N  )-mode bisymmetric Gaussian states (see Sec. 4) with respectto the  M  | N   bipartition of the modes8, as well as for ( M   + N  )-mode Gaus-sian states with fully degenerate symplectic spectrum9. In the special, butimportant case  M   = 1, PPT is a necessary and sufficient condition for sep-arability of all Gaussian states10 , 11. For a general Gaussian state of any M  | N   bipartition, the PPT criterion is replaced by another necessary andsufficient condition stating that a CM  σ  corresponds to a separable stateif and only if there exists a pair of CMs  σ A  and  σ B , relative to the sub-systems  A  and  B  respectively, such that the following inequality holds11: σ  ≥  σ A ⊕ σ B . This criterion is not very useful in practice. Alternatively,one can introduce an operational criterion based on a nonlinear map, thatis independent of (and strictly stronger than) the PPT condition12.In phase space, partial transposition amounts to a mirror reflection of one quadrature in the reduced CM of one of the parties. If   { ˜ ν  i }  is thesymplectic spectrum of the partially transposed CM ˜ σ , then a (1+ N  )-mode(or bisymmetric ( M   +  N  )-mode) Gaussian state with CM  σ  is separableif and only if ˜ ν  i  ≥  1  ∀ i . A proper measure of CV entanglement is the logarithmic negativity  13 E   N   ≡  log  ˜ ̺  1 , where   ·  1  denotes the tracenorm, which constitutes an upper bound to the  distillable entanglement   of the state  ̺ . It can be computed in terms of the symplectic spectrum ˜ ν  i  of ˜ σ : E   N   = max  0 ,  −  i :˜ ν  i < 1 log ˜ ν  i   .  (3) E   N   quantifies the extent to which the PPT condition ˜ ν  i  ≥ 1 is violated. 3. Two–Mode Gaussian States: Entanglement andMixedness Two–mode Gaussian states represent the prototypical quantum statesof CV systems, and constitute an ideal test-ground for the theoretical andexperimental investigation of CV entanglement14. Their CM can be writtenis the following block form σ  ≡   α γ γ  T  β   ,  (4)where the three 2 × 2 matrices  α ,  β ,  γ   are, respectively, the CMs of the tworeduced modes and the correlation matrix between them. It is well known10that for any two–mode CM  σ  there exists a local symplectic operation S  l  =  S  1 ⊕ S  2  which takes  σ  to its standard form  σ sf  , characterized by α  = diag { a, a } ,  β  = diag { b, b } ,  γ   = diag { c + , c − } .  (5)  February 1, 2008 10:33 WSPC/Trim Size: 9in x 6in for Review Volume AdessoIlluminati˙NEW Bipartite and Multipartite Entanglement of Gaussian States  5 States whose standard form fulfills  a  =  b  are said to be symmetric. Anypure state is symmetric and fulfills  c +  = − c −  = √  a 2 − 1. The uncertaintyprinciple Ineq. (1) can be recast as a constraint on the  Sp (4 , R )  invariantsDet σ  and ∆( σ ) = Det α + Det β +2Det γ  , yielding ∆( σ ) ≤ 1+ Det σ . Thestandard form covariances  a ,  b ,  c + , and  c −  can be determined in terms of the two local symplectic invariants µ 1  = (Det α ) − 1 / 2 = 1 /a, µ 2  = (Det β ) − 1 / 2 = 1 /b,  (6)which are the marginal purities of the reduced single–mode states, and of the two global symplectic invariants µ  = (Det σ ) − 1 / 2 = [( ab − c 2+ )( ab − c 2 − )] − 1 / 2 ,  ∆ =  a 2 + b 2 + 2 c + c − ,  (7)where  µ  is the global purity of the state. Eqs. (6-7) can be inverted to provide the following physical parametrization of two–mode states in termsof the four independent parameters  µ 1 , µ 2 , µ , and ∆15: a  = 1 µ 1 , b  = 1 µ 2 , c ±  = √  µ 1 µ 2 4  ǫ − ± ǫ +  ,  (8)with  ǫ ∓  ≡   [∆ − ( µ 1 ∓ µ 2 ) 2 / ( µ 21 µ 22 )] 2 − 4 /µ 2 . The uncertainty principleand the existence of the radicals appearing in Eq. (8) impose the followingconstraints on the four invariants in order to describe a physical state µ 1 µ 2  ≤ µ  ≤  µ 1 µ 2 µ 1 µ 2  + | µ 1 − µ 2 | ,  (9)2 µ  + ( µ 1 − µ 2 ) 2 µ 21 µ 22 ≤ ∆  ≤  1 + 1 µ 2  .  (10)The physical meaning of these constraints, and the role of the extremalstates ( i.e.  states whose invariants saturate the upper or lower bounds of Eqs. (9-10)) in relation to the entanglement, will be investigated soon. In terms of symplectic invariants, partial transposition corresponds toflipping the sign of Det γ  , so that ∆ turns into ˜∆ = ∆ − 4Det γ   =  − ∆ +2 /µ 21  + 2 /µ 22 . The symplectic eigenvalues of the CM  σ  and of its partialtranspose ˜ σ  are promptly determined in terms of symplectic invariants2 ν  2 ∓  = ∆ ∓   ∆ 2 − 4 /µ 2 ,  2˜ ν  2 ∓  = ˜∆ ∓   ˜∆ 2 − 4 /µ 2 ,  (11)where in our naming convention  ν  −  ≤  ν  +  in general, and similarly for the˜ ν  ∓ . The PPT criterion yields a state  σ  separable if and only if ˜ ν  −  ≥  1.Since ˜ ν  +  >  1 for all two–mode Gaussian states, the quantity ˜ ν  −  alsocompletely quantifies the entanglement, in fact the logarithmic negativ-ity Eq. (3) is a monotonically decreasing and convex function of ˜ ν  − , E   N   = max { 0 , − log ˜ ν  − } . In the special instance of symmetric Gaussian
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