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Informal Quantum Information Gathering 2007. Strong Monogamy and Genuine Multipartite Entanglement. The Gaussian Case. Gerardo Adesso. Quantum Theory group University of Salerno. Contents. What we know on entanglement monogamy Distributed bipartite entanglement: CKW Qubits (spin chains) - PowerPoint PPT Presentation

Strong Monogamy and Genuine Multipartite EntanglementGerardo AdessoInformal Quantum Information Gathering 2007The Gaussian Case Quantum Theory groupUniversity of Salerno

Monogamy and promiscuity of Gaussian entanglement

ContentsWhat we know on entanglement monogamyDistributed bipartite entanglement: CKWQubits (spin chains)Gaussian states (harmonic lattices)A stronger monogamy constraintAddressing shared bipartite and multipartite entanglementGenuine N-party entanglement of symmetric N-mode Gaussian statesScale invariance and molecular entanglement hierarchyBased on recently published joint works with: F. Illuminati (Salerno), A. Serafini (London), M. Ericsson (Cambridge), T. Hiroshima (Tokyo); and especially on: G. Adesso & F. Illuminati, quant-ph/0703277

Monogamy and promiscuity of Gaussian entanglement

Entanglement is monogamousSuppose that A-B and A-C are both maximally entangledthen Alice could exploit both channels simultaneously to achieve perfect 12 telecloning, violating no-cloning theoremB. M. Terhal, IBM J. Res. & Dev. 48, 71 (2004); G. Adesso & F. Illuminati, Int. J. Quant. Inf. 4, 383 (2006)

Monogamy and promiscuity of Gaussian entanglement

CKW monogamy inequalityNo-sharing for maximal entanglement, but nonmaximal one can be shared under some constraintsWhile monogamy is a fundamental quantum property, fulfillment of the above inequality depends on the entanglement measure[CKW] Coffman, Kundu, Wootters, Phys. Rev. A (2000) it was originally proven for 3 qubits using the tangle (squared concurrence)The difference between LHS and RHS yields the residual three-way shared entanglement Dr, Vidal, Cirac, Phys. Rev. A (2000)

Monogamy and promiscuity of Gaussian entanglement

Generalized monogamy inequalityConjectured by CKW (2000); proven for N qubits by Osborne and Verstraete, Phys. Rev. Lett. (2006)The difference between LHS and RHS yields not the genuine N-way shared entanglement, but all the quantum correlations not stored in couplewise form

Monogamy and promiscuity of Gaussian entanglement

Continuous variable systemsQuantum systems such as material particles (motional degrees of freedom), harmonic oscillators, light modes, or bosonic fieldsInfinite-dimensional Hilbert spaces for N modesQuadrature operators

states whose Wigner function is GaussianGaussian states can be realized experimentally with current technology (e.g. coherent, squeezed states) successfully implemented in CV quantum information processes (teleportation, QKD, )G. Adesso and F. Illuminati, review article, quant-ph/0701221, J. Phys. A in pressVector of first moments (arbitrarily adjustable by local displacements: we will set them to 0) Covariance Matrix (CM) (real, symmetric, 2N x 2N) of the second moments fully determined by

Monogamy and promiscuity of Gaussian entanglement

Monogamy of Gaussian entanglementimpliesimpliesG. Adesso and F. Illuminati, New J. Phys. 8, 15 (2006) G. Adesso, A. Serafini, and F. Illuminati, Phys. Rev. A 73, 032345 (2006)T. Hiroshima, G. Adesso, and F. Illuminati, Phys. Rev. Lett. 98, 050503 (2007)G. Adesso and F. Illuminati, review article, quant-ph/0701221, J. Phys. A in pressGaussian contangle[GCR* of (log-neg)2]

Gaussian tangle[GCR* of negativity2]

*GCR = Gaussian convex roof minimum of the average pure-state entanglement over all decompositions of the mixed Gaussian state into pure Gaussian states

Monogamy inequality for 3-mode Gaussian statesNJP 2006, PRA 2006GLOCC monotonicity of three-way residual entanglementNJP 2006---(didnt check!)Monogamy inequality for fully symmetric N-mode Gaussian statesNJP 2006Monogamy inequality for all (pure and mixed) N-mode Gaussian statesnumerical evidence

NJP 2006full analytical proofPRL 2007

Monogamy and promiscuity of Gaussian entanglement

A stronger monogamy constraintConsider a general quantum system multipartitioned in N subsystems

each comprising, in principle, one or more elementar units (qubit, mode, )

Monogamy and promiscuity of Gaussian entanglement

Decomposing the block entanglementj=2N1 2 3 4 N 1 j + +=? iterative structure

Each K-partite entanglement (K

Strong monogamy inequality fundamental requirement

implies the traditional generalized monogamy inequality (in which only the two-party entanglements are subtracted)

extremely hard to prove in general!

Monogamy and promiscuity of Gaussian entanglement

Permutation-invariant quantum statesWhy consider such instancesMain testgrounds for theoretical investigations of multipartite entanglement (structure, scaling, etc)Main practical resources for multiparty quantum information & communication protocols (teleportation networks, secret sharing, )What matters to our constructionEntanglement contributions are independent of mode indexesNo minimization required over focus partyWe can use combinatorics

Monogamy and promiscuity of Gaussian entanglement

Resolving the recursiongenuine N-party entanglementconjectured general expression for the genuine N-partite entanglement of permutation-invariant quantum states (for a proper monotone E)

alternating finite sum involving only bipartite entanglements between one party and a block of K other parties

Monogamy and promiscuity of Gaussian entanglement

Validating the conjecturevan Loock & Braunstein, Phys. Rev. Lett. (2000) G. Adesso, A. Serafini, and F. Illuminati, Phys. Rev. Lett. 93, 220504 (2004)G. Adesso and F. Illuminati, Phys. Rev. Lett. 95, 150503 (2005) in permutation-invariant Gaussian states

Monogamy and promiscuity of Gaussian entanglement

Ingredients LxK 1x1 entanglement Localized 1x1 state is a GLEMS (min-uncertainty mixed state) fully specified by global (two-mode) and local (single-mode) determinants, --.i.e. det L(N+M) , det K(N+M) , and det L+K(N+M) Gaussian entanglement measures (including contangle) are known for GLEMS G. Adesso, A. Serafini, and F. Illuminati, Phys. Rev. Lett. 93, 220504 (2004); A. Serafini, G. Adesso, and F. Illuminati, Phys. Rev. A 71, 023249 (2005).G. Adesso, A. Serafini, and F. Illuminati, Phys. Rev. Lett. 92, 087901 (2004); Phys. Rev. A 70, 022318 (2005).G. Adesso and F. Illuminati, New J. Phys. 8, 15 (2006); Phys. Rev. A 72, 032334 (2005).N-party contangle

Monogamy and promiscuity of Gaussian entanglement

N-party contanglepure states (M=0)ALWAYS POSITIVEincreases with the average squeezing rdecreases with the number of modes N(for mixed states) decreases with mixedness Mgoes to zero in the field limit (N+M)

Monogamy and promiscuity of Gaussian entanglement

Entanglement is strongly monogamous in permutation-invariant Gaussian statesvan Loock & Braunstein, Phys. Rev. Lett. (2000) G. Adesso and F. Illuminati, Phys. Rev. Lett. 95, 150503 (2005)Yonezawa, Aoki, Furusawa, Nature (2004) [experimental demonstration for N=3]Genuine N-partite entanglement is monotonic in the optimal teleportation-network fidelity (in turn, given by the localizable entanglement), and exhibits the same scaling with NTheres a unique, theoretically and operationally motivated, entanglement quantification in symmetric Gaussian states (generalizing the known cases N=2, 3)

Monogamy and promiscuity of Gaussian entanglement

Scale invariance of entanglementdet (mN)mK is independent of m Strong monogamy constrains also multipartite entanglement of molecules each made by more than one modeN-partite entanglement of N molecules (each of the same size m) is independent of m, i.e. it is scale invariant

Monogamy and promiscuity of Gaussian entanglement

Molecular entanglement hierarchy a smaller number of larger molecules share strictly more entanglement than a larger number of smaller molecules

(but all forms of multipartite entanglement are nonzero)At fixed N

Monogamy and promiscuity of Gaussian entanglement

Promiscuity Promiscuity means that bipartite and genuine multipartite entanglement are increasing functions of each other, and the genuine multipartite entanglement is enhanced by the simultaneous presence of the bipartite one, while typically in low-dimensional systems like qubits only the opposite behaviour is compatible with monogamy (cfr. GHZ and W states of qubits) This was pointed out in our first Gaussian monogamy paper (NJP 2006) in the subcase N=3 of permutation-invariant Gaussian states, there re-baptised CV GHZ/W states

: the general pictureMichael PerezCreative threesomein N-mode permutation-invariant Gaussian states, all possible forms of bipartite and multipartite entanglement coexist and are mutually enhanced for any nonzero squeezingunlimited promiscuity also occurs in some four-mode nonsymmetric Gaussian states, where strong monogamy can be proven to hold as well Despite entanglement is strongly monogamous, theres infinitely more freedom when addressing distributed correlations in harmonic CV systems, as compared to qubit systems

Monogamy and promiscuity of Gaussian entanglement

Summary and concluding remarksWe recalled the current state-of-the-art on entanglement sharing including our recent conclusive results on monogamy of Gaussian statesWe proposed a novel, general approach to monogamy, wherein a stronger a priori constraint imposes trade-offs on both bipartite and multipartite entanglement on the same groundWe demonstrated that approach to be successful when dealing with permutation-invariant Gaussian states of an arbitrary N-mode continuous variable system: namely, I derived an analytic expression for the genuine N-pa