Strong Monogamy and Genuine Multipartite Entanglement

1. Informal Quantum Information Gathering2007 Strong Monogamy and Genuine Multipartite Entanglement The Gaussian Case Gerardo Adesso Quantum Theory group University of Salerno

2. Contents • What we know on entanglement monogamy • Distributed bipartite entanglement: CKW • Qubits (spin chains) • Gaussian states (harmonic lattices) • 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 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 Strong Monogamy and Genuine Multipartite Entanglement

3. Entanglement is monogamous Bob Charlie Alice Suppose that A-B and A-C are both maximally entangled… …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) Strong Monogamy and Genuine Multipartite Entanglement

4. 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 • 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 [CKW] Coffman, Kundu, Wootters, Phys. Rev. A (2000) Dür, Vidal, Cirac, Phys. Rev. A (2000) Strong Monogamy and Genuine Multipartite Entanglement

5. ‘Generalized’ monogamy inequality … … ≥ … + … … + + The difference between LHS and RHS yields not the genuine N-way shared entanglement, but all the quantum correlations not stored in couplewise form Conjectured by CKW (2000); proven for N qubits by Osborne and Verstraete, Phys. Rev. Lett. (2006) Strong Monogamy and Genuine Multipartite Entanglement

6. 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 Gaussian states  states whose Wigner function is Gaussian • can be realized experimentally with current technology (e.g. coherent, squeezed states) • successfully implemented in CV quantum information processes (teleportation, QKD, …) • Vector of first moments (arbitrarily adjustable by local displacements: we will set them to 0) • Covariance Matrix (CM)σ(real, symmetric, 2N x2N) of the second moments • fully determined by G. Adesso and F. Illuminati, review article, quant-ph/0701221, J. Phys. A in press Strong Monogamy and Genuine Multipartite Entanglement

7. traditional Monogamy of Gaussian entanglement Gaussian contangle [GCR* of (log-neg)2] Gaussian tangle [GCR* of negativity2] implies implies G. Adesso and F. Illuminati, New J. Phys. 8, 15 (2006) G. Adesso, A. Serafini, and F. Illuminati, Phys. Rev. A73, 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 *GCR = “Gaussian convex roof” minimum of the average pure-state entanglement over all decompositions of the mixed Gaussian state into pure Gaussian states Strong Monogamy and Genuine Multipartite Entanglement

8. A stronger monogamy constraint… • Does a stronger, general monogamy inequalitya 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? Consider a general quantum system multipartitioned in N subsystems each comprising, in principle, one or more elementar units (qubit, mode, …) Strong Monogamy and Genuine Multipartite Entanglement

9. Decomposing the block entanglement N 1 jk ∑ + k>j 1 2 3 4 N … Genuine N-partite entanglement min … = focus p1|p2|…|pN E 1 2 3 4 N ? N 1 j ∑ ≥ … = j=2 +··· + • 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 Strong Monogamy and Genuine Multipartite Entanglement

10. Strong monogamy inequality ? p1|p2|…|pN ≥ 0 E • fundamental requirement • implies the traditional ‘generalized’ monogamy inequality (in which only the two-party entanglements are subtracted) • extremely hard to prove in general! Strong Monogamy and Genuine Multipartite Entanglement

11. Permutation-invariant quantum states 2 2 N=3 genuine N-party entanglement N=4 3 3 N-1 N=any (N-1) K N-1 K • 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 Strong Monogamy and Genuine Multipartite Entanglement

12. Resolving the recursion 2 2 N=3 N=4 3 3 N-1 N=any (N-1) K N-1 K 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 genuine N-party entanglement Strong Monogamy and Genuine Multipartite Entanglement

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

14. Ingredients N-party contangle L • Unitary localizability ULUK L+K-2 K CM σ(N)= z z b L L z b L z z z b L • 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 σL(N+M) , det σK(N+M) , and det σL+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. A71, 023249 (2005). G. Adesso, A. Serafini, and F. Illuminati, Phys. Rev. Lett.92, 087901 (2004); Phys. Rev. A70, 022318 (2005). G. Adesso and F. Illuminati, New J. Phys. 8, 15 (2006); Phys. Rev. A72, 032334 (2005). Strong Monogamy and Genuine Multipartite Entanglement

15. 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) Strong Monogamy and Genuine Multipartite Entanglement

16. Entanglement is strongly monogamous… … in permutation-invariant Gaussian states 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 There’s a unique, theoretically and operationally motivated, entanglement quantification in symmetric Gaussian states (generalizing the known cases N=2, 3) 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] Strong Monogamy and Genuine Multipartite Entanglement

17. Scale invariance of entanglement = = det σ(mN)mK is independent of m … • Strong monogamy constrains also multipartite entanglement of moleculeseach 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 Strong Monogamy and Genuine Multipartite Entanglement

18. Molecular entanglement hierarchy 0.4 G r e s ¹ r 0.3 ¿ 0.2 0.1 0 0.2 0.4 0.6 0.8 1 (N=12) At fixed N… • …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…) Strong Monogamy and Genuine Multipartite Entanglement

19. Promiscuity : the general picture Michael Perez Fire • ‘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 Michael Perez Creative threesome Despite entanglement is strongly monogamous, there’s infinitely more freedom when addressing distributed correlations in harmonic CV systems, as compared to qubit systems 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…  Strong Monogamy and Genuine Multipartite Entanglement

20. 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 Strong Monogamy and Genuine Multipartite Entanglement

21. 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 Strong Monogamy and Genuine Multipartite Entanglement

22. Thank you for the attention! Strong Monogamy and Genuine Multipartite Entanglement

23. Strong Monogamy and Genuine Multipartite Entanglement