Distribution of entanglement and correlations in all finite dimensions

Christopher Eltschka; Jens Siewert

doi:10.22331/q-2018-05-22-64

Distribution of entanglement and correlations in all finite dimensions

Christopher Eltschka¹ and Jens Siewert^2,3

¹Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany
²Departamento de Química Física, Universidad del País Vasco UPV/EHU, E-48080 Bilbao, Spain
³IKERBASQUE Basque Foundation for Science, E-48013 Bilbao, Spain

Published:	2018-05-22, volume 2, page 64
Eprint:	arXiv:1708.09639v2
Doi:	https://doi.org/10.22331/q-2018-05-22-64
Citation:	Quantum 2, 64 (2018).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

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

@article{Eltschka2018distributionof,
  doi = {10.22331/q-2018-05-22-64},
  url = {https://doi.org/10.22331/q-2018-05-22-64},
  title = {Distribution of entanglement and correlations in all finite dimensions},
  author = {Eltschka, Christopher and Siewert, Jens},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {2},
  pages = {64},
  month = may,
  year = {2018}
}

► 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

[1] Christopher Eltschka and Jens Siewert, "Joint Schmidt-type decomposition for two bipartite pure quantum states", Physical Review A 101 2, 022302 (2020).

[2] Sebastian Gartzke and Andreas Osterloh, "Generalized W state of four qubits with exclusively the three-tangle", arXiv:1712.08595, Physical Review A 98 5, 052307 (2018).

[3] Nikolai Wyderka, Felix Huber, and Otfried Gühne, "Constraints on correlations in multiqubit systems", Physical Review A 97 6, 060101 (2018).

[4] Christopher Eltschka and Jens Siewert, "MaximumN-body correlations do not in general imply genuine multipartite entanglement", Quantum 4, 229 (2020).

[5] Satoya Imai, Nikolai Wyderka, Andreas Ketterer, and Otfried Gühne, "Bound Entanglement from Randomized Measurements", Physical Review Letters 126 15, 150501 (2021).

[6] Paul Appel, Marcus Huber, and Claude Klöckl, "Monogamy of correlations and entropy inequalities in the Bloch picture", arXiv:1710.02473, Journal of Physics Communications 4 2, 025009 (2020).

[7] Jie Zhu, Meng-Jun Hu, Yue Dai, Yan-Kui Bai, S. Camalet, Chengjie Zhang, Chuan-Feng Li, Guang-Can Guo, and Yong-Sheng Zhang, "Realization of the tradeoff between internal and external entanglement", Physical Review Research 2 4, 043068 (2020).

[8] Felix Huber, "Positive maps and trace polynomials from the symmetric group", Journal of Mathematical Physics 62 2, 022203 (2021).

[9] Waldemar Kłobus, Wiesław Laskowski, Tomasz Paterek, Marcin Wieśniak, and Harald Weinfurter, "Higher dimensional entanglement without correlations", The European Physical Journal D 73 2, 29 (2019).

[10] N Wyderka and O Gühne, "Characterizing quantum states via sector lengths", Journal of Physics A: Mathematical and Theoretical 53 34, 345302 (2020).

[11] Christopher Eltschka, Felix Huber, Otfried Gühne, and Jens Siewert, "Exponentially many entanglement and correlation constraints for multipartite quantum states", Physical Review A 98 5, 052317 (2018).

[12] Yu Guo and Lin Zhang, "Multipartite entanglement measure and complete monogamy relation", Physical Review A 101 3, 032301 (2020).

[13] Xinwei Zha, Irfan Ahmed, Da Zhang, and Yanpeng Zhang, "Generalized monogamy linear entropy relations for multi-qubit pure states", Laser Physics 30 3, 035201 (2020).

The above citations are from Crossref's cited-by service (last updated successfully 2021-04-21 22:43:56) and SAO/NASA ADS (last updated successfully 2021-04-21 22:43:57). The list may be incomplete as not all publishers provide suitable and complete 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.