The physics of a many-particle system is determined by the correlations in its quantum state. Therefore, analyzing these correlations is the foremost task of many-body physics. Any 'a priori' constraint for the properties of the global vs. the local states-the so-called marginals-would help in order to narrow down the wealth of possible solutions for a given many-body problem, however, little is known about such constraints. We derive an equality for correlation-related quantities of any multipartite quantum system composed of finite-dimensional local parties. This relation defines a necessary condition for the compatibility of the marginal properties with those of the joint state. While the equality holds both for pure and mixed states, the pure-state version containing only entanglement measures represents a fully general monogamy relation for entanglement. These findings have interesting implications in terms of conservation laws for correlations, and also with respect to topology.
► BibTeX data
► References
[1] E. Schrödinger, Die gegenwärtige Situation in der Quantenmechanik, Naturwissenschaften 23 (49), 53 (1935).
https://doi.org/10.1007/BF01491891
[2] A. Peres, Quantum Theory: Concepts and Methods, (Kluwer Academic Publishers, New York, 2002).
[3] H.M. Wiseman, S.J. Jones, and A.C. Doherty, Steering, Entanglement, Nonlocality, and the Einstein-Podolsky-Rosen Paradox, Phys. Rev. Lett. 98, 140402 (2007).
https://doi.org/10.1103/PhysRevLett.98.140402
[4] R.F. Werner, Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model, Phys. Rev. A 40, 4277 (1989).
https://doi.org/10.1103/PhysRevA.40.4277
[5] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell nonlocality, Rev. Mod. Phys. 86, 419 (2014).
https://doi.org/10.1103/RevModPhys.86.419
[6] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009).
https://doi.org/10.1103/RevModPhys.81.865
[7] C. Eltschka and J. Siewert, Quantifying entanglement resources, J. Phys. A: Math. Theor. 47, 424005 (2014).
https://doi.org/10.1088/1751-8113/47/42/424005
[8] A. Streltsov, Quantum Correlations Beyond Entanglement, SpringerBriefs in Physics, (Springer International Publishing, 2015).
[9] G. Adesso, C.R. Bromley, and M. Cianciaruso, Measures and applications of quantum correlations, J. Phys. A: Math. Theor. 49, 473001 (2016).
https://doi.org/10.1088/1751-8113/49/47/473001
[10] V. Coffman, J. Kundu, and W.K. Wootters, Distributed entanglement, Phys. Rev. A 61, 052306 (2000).
https://doi.org/10.1103/PhysRevA.61.052306
[11] M. Koashi and A. Winter, Monogamy of entanglement and other correlations, Phys. Rev. A 69, 022309 (2004).
https://doi.org/10.1103/PhysRevA.69.022309
[12] T.J. Osborne and F. Verstraete, General Monogamy Inequality for Bipartite Qubit Entanglement, Phys. Rev. Lett. 96, 220503, (2006).
https://doi.org/10.1103/PhysRevLett.96.220503
[13] Y.-K. Bai, Y.-F. Xu, and Z.D. Wang, General Monogamy Relation for the Entanglement of Formation in Multiqubit Systems, Phys. Rev. Lett. 113, 100503 (2014).
https://doi.org/10.1103/PhysRevLett.113.100503
[14] B. Regula, S. Di Martino, S.-J. Lee, and G. Adesso, Strong monogamy conjecture for multiqubit entanglement: The four-qubit case, Phys. Rev. Lett. 113, 110501 (2014).
https://doi.org/10.1103/PhysRevLett.113.110501
[15] H.S. Dhar, A. Kumar Pal, D. Rakshit, A. Sen De, and U. Sen, Monogamy of quantum correlations - a review, Lectures on General Quantum Correlations and their Applications, Springer International Publishing, 23 (2017).
https://doi.org/10.1007/978-3-319-53412-1
[16] B. Toner, Monogamy of nonlocal correlations, Proc. R. Soc. A 465, 59 (2009).
https://doi.org/10.1098/rspa.2008.0149
[17] M.P. Seevinck, Monogamy of Correlations vs. Monogamy of Entanglement Quant. Inf. Proc. 9, 273 (2010).
https://doi.org/10.1007/s11128-009-0161-6
[18] C. Eltschka and J. Siewert, Monogamy equalities for qubit entanglement from Lorentz invariance, Phys. Rev. Lett. 114, 140402 (2015).
https://doi.org/10.1103/PhysRevLett.114.140402
[19] A.A. Klyachko, Quantum marginal problem and N-representability, J. Phys.: Conf. Ser. 36, 72 (2006).
https://doi.org/10.1088/1742-6596/36/1/014
[20] A. Wong and N. Christensen, Potential multiparticle entanglement measure, Phys. Rev. A 63, 044301 (2001).
https://doi.org/10.1103/PhysRevA.63.044301
[21] M. Horodecki and P. Horodecki, Reduction criterion of separability and limits for a class of distillation protocols, Phys. Rev. A 59, 4206 (1999).
https://doi.org/10.1103/PhysRevA.59.4206
[22] P. Rungta, V. Buzek, C.M. Caves, M. Hillery, and G.J. Milburn, Universal state inversion and concurrence in arbitrary dimensions, Phys. Rev. A 64, 042315 (2001).
https://doi.org/10.1103/PhysRevA.64.042315
[23] W. Hall, Multipartite reduction criteria for separability, Phys. Rev. A 72, 022311 (2005).
https://doi.org/10.1103/PhysRevA.72.022311
[24] W. Hall, A new criterion for indecomposability of positive maps, J. Phys. A: Math. Gen. 39, 14119 (2006).
https://doi.org/10.1088/0305-4470/39/45/020
[25] H.-P. Breuer, Optimal Entanglement Criterion for Mixed Quantum States, Phys. Rev. Lett. 97, 080501 (2006).
https://doi.org/10.1103/PhysRevLett.97.080501
[26] M. Lewenstein, R. Augusiak, D. Chruściński, S. Rana, and J. Samsonowicz, Sufficient separability criteria and linear maps, Phys. Rev. A 93, 042335 (2016).
https://doi.org/10.1103/PhysRevA.93.042335
[27] W.K. Wootters, Entanglement of Formation of an Arbitrary State of Two Qubits, Phys. Rev. Lett. 80, 2245 (1998).
https://doi.org/10.1103/PhysRevLett.80.2245
[28] H. Georgi, Lie algebras in particle physics, Frontiers in physics, vol. 54. (Addison-Wesley, Redwood City, 1982).
[29] S. Albeverio and S.-M. Fei, A note on invariants and entanglements, J. Opt. B 3, 223 (2001).
https://doi.org/10.1088/1464-4266/3/4/305
[30] S.J. Akhtarshenas, Concurrence vectors in arbitrary multipartite quantum systems, J. Phys. A: Math. Gen. 38, 6777 (2005).
https://doi.org/10.1088/0305-4470/38/30/011
[31] Y.-Q. Li and G.-Q. Zhu, Concurrence vectors for entanglement of high-dimensional systems, Front. Phys. China 3, 250 (2008).
https://doi.org/10.1007/s11467-008-0022-2
[32] G. Vidal, Entanglement monotones, J. Mod. Opt. 47, 355 (2000).
https://doi.org/10.1080/09500340008244048
[33] W. Dür, G. Vidal, and J.I. Cirac, Three qubits can be entangled in two different ways, Phys. Rev. A 62, 062314 (2000).
https://doi.org/10.1103/PhysRevA.62.062314
[34] A. Uhlmann, Roofs and convexity, Entropy 12, 1799 (2010).
https://doi.org/10.3390/e12071799
[35] C. Eltschka, T. Bastin, A. Osterloh, and J. Siewert, Multipartite-entanglement monotones and polynomial invariants, Phys. Rev. A 85, 022301, (2012); Erratum, ibid., 059903 (2012).
https://doi.org/10.1103/PhysRevA.85.022301
[36] J.-M. Cai, Z.-W. Zhou, S. Zhang, and G.-C. Guo, Compatibility conditions from multipartite entanglement measures, Phys. Rev. A 75, 052324 (2007).
https://doi.org/10.1103/PhysRevA.75.052324
[37] E.H. Lieb and M.B. Ruskai, Proof of the strong subadditivity of quantum-mechanical entropy, J. Math. Phys. 14, 1938 (1973).
https://doi.org/10.1063/1.1666274
[38] M.D. Crossley, Essential topology, Springer Undergraduate Mathematics Series (Springer, London, 2005).
► Cited by (beta)
Crossref's cited-by service has no data on citing works. Unfortunately not all publishers provide suitable citation data.
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.