1. Strong Monogamy and Genuine Multipartite Entanglement Gerardo Adesso Informal Quantum Information Gathering 2007 The Gaussian Case Quantum Theory group University of Salerno

2. G. AdessoStrong Monogamy and Genuine Multipartite Entanglement2 Contents What we know on entanglement monogamy What we know on entanglement monogamy Distributed bipartite entanglement: CKW Qubits (spin chains) Gaussian states (harmonic lattices) A stronger monogamy constraint… A stronger monogamy constraint… Addressing shared bipartite and multipartite entanglement Genuine N-party entanglement of symmetric N-mode Gaussian states Scale invariance and molecular entanglement hierarchy Scale invariance and molecular entanglement hierarchy Based 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

3. G. AdessoStrong Monogamy and Genuine Multipartite Entanglement3 Entanglement is monogamous Suppose that A-B and A-C are both maximally entangled… Alice BobCharlie …then Alice could exploit both channels simultaneously to achieve perfect 1 2 telecloning, violating no-cloning theorem B. M. Terhal, IBM J. Res. & Dev. 48, 71 (2004); G. Adesso & F. Illuminati, Int. J. Quant. Inf. 4, 383 (2006)

4. G. AdessoStrong Monogamy and Genuine Multipartite Entanglement4 CKW monogamy inequality No-sharing for maximal entanglement, but nonmaximal one can be shared… under some constraints While 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 Dür, Vidal, Cirac, Phys. Rev. A (2000)

5. G. AdessoStrong Monogamy and Genuine Multipartite Entanglement5 Generalized monogamy inequality … … … … + … + Conjectured 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

6. G. AdessoStrong Monogamy and Genuine Multipartite Entanglement6 Continuous variable systems Quantum systems such as material particles (motional degrees of freedom), harmonic oscillators, light modes, or bosonic fields Infinite-dimensional Hilbert spaces for N modes Quadrature operators states whose Wigner function is Gaussian Gaussian 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 press Vector of first moments (arbitrarily adjustable by local displacements: we will set them to 0) Covariance Matrix (CM) σ (real, symmetric, 2N x 2N) of the second moments fully determined by

7. G. AdessoStrong Monogamy and Genuine Multipartite Entanglement7 Monogamy of Gaussian entanglement Monogamy inequality for 3-mode Gaussian states NJP 2006, PRA 2006 GLOCC monotonicity of three-way residual entanglement NJP 2006 --- (didnt check!) Monogamy inequality for fully symmetric N-mode Gaussian states NJP 2006 Monogamy inequality for all (pure and mixed) N- mode Gaussian states numerical evidence NJP 2006 full analytical proof PRL 2007 implies G. 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 press Gaussian contangle [GCR* of (log-neg) 2 ] Gaussian tangle [GCR* of negativity 2 ] traditional *GCR = Gaussian convex roof minimum of the average pure-state entanglement over all decompositions of the mixed Gaussian state into pure Gaussian states

8. G. AdessoStrong Monogamy and Genuine Multipartite Entanglement8 A stronger monogamy constraint… Consider a general quantum system multipartitioned in N subsystems each comprising, in principle, one or more elementar units (qubit, mode, …) Does a stronger, general monogamy inequality a priori dictated by quantum mechanics exist, which constrains on the same ground the distribution of bipartite and multipartite entanglement? Does a stronger, general monogamy inequality a priori dictated by quantum mechanics exist, which constrains on the same ground the distribution of bipartite and multipartite entanglement? Can we provide a suitable generalization of the tripartite analysis to arbitrary N, such that a genuine N- partite entanglement quantifier is naturally derived? Can we provide a suitable generalization of the tripartite analysis to arbitrary N, such that a genuine N- partite entanglement quantifier is naturally derived?

9. G. AdessoStrong Monogamy and Genuine Multipartite Entanglement9 Decomposing the block entanglement … j=2 N 1 2 3 4 N 1j + k>j N 1 j k +··· +1 2 3 4 N … = ? iterative structure Each K-partite entanglement (K<N) is obtained by nesting the same decomposition at orders 3,…,K The N-partite entanglement is then implicitly defined by difference … Genuine N-partite N-partite entanglement entanglement = min focus E p 1 |p 2 |…|p N

10. G. AdessoStrong Monogamy and Genuine Multipartite Entanglement10 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! E p 1 |p 2 |…|p N 0 ?

11. G. AdessoStrong Monogamy and Genuine Multipartite Entanglement11 Permutation-invariant quantum states Why consider such instances Main 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 construction Entanglement contributions are independent of mode indexes No minimization required over focus party We can use combinatorics (N-1) N-1 K K N=any 33 N=4 22 N=3 genuine N-party entanglement

12. G. AdessoStrong Monogamy and Genuine Multipartite Entanglement12 Resolving the recursion (N-1) N-1 K K N=any 33 N=4 22 N=3 genuine N-party entanglement conjectured 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

13. G. AdessoStrong Monogamy and Genuine Multipartite Entanglement13 van 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) CM σ (N) = General instance (one mode per party) mixed N-mode state obtained from pure (N+M)-mode state by tracing over M modes N M Validating the conjecture… … in permutation-invariant Gaussian states

14. G. AdessoStrong Monogamy and Genuine Multipartite Entanglement14 Ingredients Unitary localizability L K U L U K L+K-2 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 σ L (N+M), det σ K (N+M), and det σ 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 CM σ (N) =

15. G. AdessoStrong Monogamy and Genuine Multipartite Entanglement15 N-party contangle pure states (M=0) ALWAYS POSITIVE increases with the average squeezing r decreases with the number of modes N (for mixed states) decreases with mixedness M goes to zero in the field limit (N+M)

16. G. AdessoStrong Monogamy and Genuine Multipartite Entanglement16 Entanglement is strongly monogamous… … in permutation-invariant Gaussian states van 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 N Theres a unique, theoretically and operationally motivated, entanglement quantification in symmetric Gaussian states (generalizing the known cases N=2, 3)

17. G. AdessoStrong Monogamy and Genuine Multipartite Entanglement17 Scale invariance of entanglement det σ (mN) mK is independent of m … Strong monogamy constrains also multipartite entanglement of molecules each made by more than one mode N-partite entanglement of N molecules (each of the same size m) is independent of m, i.e. it is scale invariant ==

18. G. AdessoStrong Monogamy and Genuine Multipartite Entanglement18 …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…) Molecular entanglement hierarchy At fixed N… 0.20.40.60.81 0.1 0.2 0.3 0.4 0 G res ¿ ¹ r (N=12)

19. G. AdessoStrong Monogamy and Genuine Multipartite Entanglement19 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 picture Michael Perez Creative threesome Michael Perez Creative threesome Michael Perez Fire Michael Perez Fire in N-mode permutation-invariant Gaussian states, all possible forms of bipartite and multipartite entanglement coexist and are mutually enhanced for any nonzero squeezing unlimited 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

20. G. AdessoStrong Monogamy and Genuine Multipartite Entanglement20 Summary and concluding remarks We recalled the current state-of-the-art on entanglement sharing including our recent conclusive results on monogamy of Gaussian states We 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 ground We 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-partite entanglement of such states, and operationally connected it to the optimal fidelity of teleportation networks employing such resources Scale invariance of continuous variable entanglement has been demonstrated in symmetric Gaussian states, and a hierarchy of molecular entanglements has been established, from which the promiscuous structure of distributed entanglement arises naturally G. Adesso and F. Illuminati, quant-ph/0703277

21. G. AdessoStrong Monogamy and Genuine Multipartite Entanglement21 Some interesting open issues Checking the strong monogamy in systems of 4 or more qubits (work in progress with M. Piani), and/or trying to prove strong monogamy in general for states of arbitrary dimension (e.g. using the squashed entanglement) Further investigating interdependence relationships between monogamy constraints, entanglement frustration effects and De Finetti-like bounds (for symmetric states) Investigating the origin of promiscuity, e.g. by considering dimension- dependent qudit families and studying the onset of shareability towards the continuous variable limit Devising new protocols to take advantage of promiscuous entanglement for communication and computation purposes

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