Entanglement marginal problems

Miguel Navascués1, Flavio Baccari2, and Antonio Acín3,4

1Institute for Quantum Optics and Quantum Information (IQOQI) Vienna, Austrian Academy of Sciences
2Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany
3ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
4ICREA-Institucio Catalana de Recerca i Estudis Avançats, Lluis Companys 23, 08010 Barcelona, Spain

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


We consider the entanglement marginal problem, which consists of deciding whether a number of reduced density matrices are compatible with an overall separable quantum state. To tackle this problem, we propose hierarchies of semidefinite programming relaxations of the set of quantum state marginals admitting a fully separable extension. We connect the completeness of each hierarchy to the resolution of an analog classical marginal problem and thus identify relevant experimental situations where the hierarchies are complete. For finitely many parties on a star configuration or a chain, we find that we can achieve an arbitrarily good approximation to the set of nearest-neighbour marginals of separable states with a time (space) complexity polynomial (linear) on the system size. Our results even extend to infinite systems, such as translation-invariant systems in 1D, as well as higher spatial dimensions with extra symmetries.

For systems composed of two or more parts, quantum theory predicts the existence of physical states that cannot be prepared by acting on each part locally in a coordinated way. Such states are said to be entangled; otherwise, they are called separable.

Determining if the state of a multi-partite quantum system is separable or entangled is a difficult task. First, acquiring a full state description of an n-partite system requires a number of experiments exponential in n. Second, even if we had such a description, the computational resources required to run general algorithms for entanglement detection scale terribly with the system size.

In this paper, we provide general methods to decide whether the near-neighbor statistics of a many-body quantum state, which can be estimated through a small number of experiments, are compatible with the existence of an overall separable state. Our methods provide an efficient characterization of quantum entanglement in physical systems where the parts or sites are arranged in 1D and tree-like geometries. We also show how to detect entanglement in higher dimensional scenarios subject to global symmetries.

► BibTeX data

► References

[1] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. S. L. Brandao, D. A. Buell, B. Burkett, Y. Chen, Z. Chen, B. Chiaro, R. Collins, W. Courtney, A. Dunsworth, E. Farhi, B. Foxen, A. Fowler, C. Gidney, M. Giustina, R. Graff, K. Guerin, S. Habegger, M. P. Harrigan, M. J. Hartmann, A. Ho, M. Hoffmann, T. Huang, T. S. Humble, S. V. Isakov, E. Jeffrey, Z. Jiang, D. Kafri, K. Kechedzhi, J. Kelly, P. V. Klimov, S. Knysh, A. Korotkov, F. Kostritsa, D. Landhuis, M. Lindmark, E. Lucero, D. Lyakh, S. Mandrà, J. R. McClean, M. McEwen, A. Megrant, X. Mi, K. Michielsen, M. Mohseni, J. Mutus, O. Naaman, M. Neeley, C. Neill, M. Y. Niu, E. Ostby, A. Petukhov, J. C. Platt, C. Quintana, E. G. Rieffel, P. Roushan, N. C. Rubin, D. Sank, K. J. Satzinger, V. Smelyanskiy, K. J. Sung, M. D. Trevithick, A. Vainsencher, B. Villalonga, T. White, Z. J. Yao, P. Yeh, A. Zalcman, H. Neven, and J. M. Martinis, Nature 574, 505 (2019).

[2] R. Schmied, J.-D. Bancal, B. Allard, M. Fadel, V. Scarani, P. Treutlein, and N. Sangouard, Science 352, 441 (2016).

[3] A. C. Doherty, P. A. Parrilo, and F. M. Spedalieri, Phys. Rev. Lett. 88, 187904 (2002).

[4] O. Gühne, M. Reimpell, and R. F. Werner, Phys. Rev. A 77, 052317 (2008).

[5] G. Tóth, C. Knapp, O. Gühne, and H. J. Briegel, Physical Review Letters 99 (2007), 10.1103/​PhysRevLett.99.250405.

[6] G. Vitagliano, I. Apellaniz, I. L. Egusquiza, and G. Tóth, Physical Review A 89 (2014), 10.1103/​PhysRevA.89.032307.

[7] J. Tura, G. De las Cuevas, R. Augusiak, M. Lewenstein, A. Acín, and J. I. Cirac, Physical Review X 7, 021005 (2017).

[8] A. Aloy, J. Tura, F. Baccari, A. Acín, M. Lewenstein, and R. Augusiak, Physical Review Letters 123 (2019), 10.1103/​physrevlett.123.100507.

[9] L. Vandenberghe and S. Boyd, SIAM Review 38, 49 (1996).

[10] S. Goldstein, T. Kuna, J. L. Lebowitz, and E. R. Speer, Journal of Statistical Physics 166, 765 (2017).

[11] A. G. Schlijper, Journal of Statistical Physics 50, 689 (1988).

[12] Z. Wang, S. Singh, and M. Navascués, Phys. Rev. Lett. 118, 230401 (2017).

[13] I. Pitowsky, Quantum Probability and Quantum Logic, Lecture Notes in Physics (Springer Berlin Heidelberg, 1989).

[14] A. C. Doherty, P. A. Parrilo, and F. M. Spedalieri, Phys. Rev. A 69, 022308 (2004).

[15] A. C. Doherty, P. A. Parrilo, and F. M. Spedalieri, Phys. Rev. A 71, 032333 (2005).

[16] S. Peres, Phys. Rev. Lett. 77, 1413–1415 (1996).

[17] M. Navascués, M. Owari, and M. B. Plenio, Phys. Rev. A 80, 052306 (2009).

[18] M. Abramowitz, Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables (Dover Publications, Inc., USA, 1974).

[19] L. Vandenberghe and M. S. Andersen, Foundations and Trends in Optimization 1, 241 (2015).

[20] J. B. Lasserre, in Mathematical Software - ICMS 2006, edited by A. Iglesias and N. Takayama (Springer Berlin Heidelberg, Berlin, Heidelberg, 2006) pp. 263–272.

[21] H. L. Bodlaender, J. R. Gilbert, H. Hafsteinsson, and T. Kloks, J. Algorithms 18, 238–255 (1995).

[22] H. L. Bodlaender, SIAM Journal on Computing 25, 1305 (1996).

[23] H. L. Bodlaender, Theoretical Computer Science 209, 1 (1998).

[24] J. Löfberg, in Proceedings of the CACSD Conference (Taipei, Taiwan, 2004).

[25] L. Vandenberghe and S. Boyd, The MOSEK optimization toolbox for MATLAB manual. Version 7.0 (Revision 140). (MOSEK ApS, Denmark.).

[26] M. Horodecki, P. Horodecki, and R. Horodecki, Physics Letters A 223, 1–8 (1996).

[27] M. Paraschiv, N. Miklin, T. Moroder, and O. Gühne, Phys. Rev. A 98, 062102 (2018).

[28] N. Miklin, T. Moroder, and O. Gühne, Phys. Rev. A 93, 020104 (2016).

[29] E. Barouch and B. M. McCoy, Physical Review A 3, 786 (1971).

[30] G. Tóth, Phys. Rev. A 71, 010301 (2005).

[31] M. R. Dowling, A. C. Doherty, and S. D. Bartlett, Phys. Rev. A 70, 062113 (2004).

[32] C. Brukner and V. Vedral, arXiv preprint quant-ph/​0406040 (2004).

[33] J. Hide, W. Son, I. Lawrie, and V. Vedral, Phys. Rev. A 76, 022319 (2007).

[34] Y. Nakata, D. Markham, and M. Murao, Phys. Rev. A 79, 042313 (2009).

[35] P. Hauke, M. Heyl, L. Tagliacozzo, and P. Zoller, Nature Physics 12, 778 (2016).

[36] I. Bose and A. Tribedi, Phys. Rev. A 72, 022314 (2005).

[37] N. Schuch and J. I. Cirac, Phys. Rev. A 82, 012314 (2010).

[38] Z. Wang and M. Navascués, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474, 20170822 (2018).

[39] J. R. S. Blair and B. Peyton, in Graph Theory and Sparse Matrix Computation, edited by A. George, J. R. Gilbert, and J. W. H. Liu (Springer New York, New York, NY, 1993) pp. 1–29.

Cited by

[1] Chung-Yun Hsieh, Matteo Lostaglio, and Antonio Acín, "Quantum channel marginal problem", Physical Review Research 4 1, 013249 (2022).

[2] Irénée Frérot, Flavio Baccari, and Antonio Acín, "Unveiling Quantum Entanglement in Many-Body Systems from Partial Information", PRX Quantum 3 1, 010342 (2022).

[3] Albert Aloy, Matteo Fadel, and Jordi Tura, "The quantum marginal problem for symmetric states: applications to variational optimization, nonlocality and self-testing", New Journal of Physics 23 3, 033026 (2021).

[4] Irénée Frérot and Tommaso Roscilde, "Optimal Entanglement Witnesses: A Scalable Data-Driven Approach", Physical Review Letters 127 4, 040401 (2021).

[5] Viktor Nordgren, Olga Leskovjanová, Jan Provazník, Natalia Korolkova, and Ladislav Mišta, "Convicting emergent multipartite entanglement with evidence from a partially blind witness", arXiv:2103.07327.

The above citations are from Crossref's cited-by service (last updated successfully 2022-05-17 11:52:10) and SAO/NASA ADS (last updated successfully 2022-05-17 11:52:11). The list may be incomplete as not all publishers provide suitable and complete citation data.