Entanglement marginal problems

Miguel Navascués; Flavio Baccari; Antonio Acín

doi:10.22331/q-2021-11-25-589

Entanglement marginal problems

Miguel Navascués¹, Flavio Baccari², and Antonio Acín^3,4

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

Published:	2021-11-25, volume 5, page 589
Eprint:	arXiv:2006.09064v5
Doi:	https://doi.org/10.22331/q-2021-11-25-589
Citation:	Quantum 5, 589 (2021).

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Abstract

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.

Popular summary

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

@article{Navascues2021entanglement,
  doi = {10.22331/q-2021-11-25-589},
  url = {https://doi.org/10.22331/q-2021-11-25-589},
  title = {Entanglement marginal problems},
  author = {Navascu{\'{e}}s, Miguel and Baccari, Flavio and Ac{\'{i}}n, Antonio},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {589},
  month = nov,
  year = {2021}
}

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Cited by

[1] Ilya Kull, Norbert Schuch, Ben Dive, and Miguel Navascués, "Lower Bounds on Ground-State Energies of Local Hamiltonians through the Renormalization Group", Physical Review X 14 2, 021008 (2024).

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

[3] Costantino Budroni, Trails in Modern Theoretical and Mathematical Physics 93 (2023) ISBN:978-3-031-44987-1.

[4] Irénée Frérot, Matteo Fadel, and Maciej Lewenstein, "Probing quantum correlations in many-body systems: a review of scalable methods", Reports on Progress in Physics 86 11, 114001 (2023).

[5] Gelo Noel M. Tabia, Kai-Siang Chen, Chung-Yun Hsieh, Yu-Chun Yin, and Yeong-Cherng Liang, "Entanglement transitivity problems", npj Quantum Information 8 1, 98 (2022).

[6] Viktor Nordgren, Olga Leskovjanová, Jan Provazník, Adam Johnston, Natalia Korolkova, and Ladislav Mišta, "Certifying emergent genuine multipartite entanglement with a partially blind witness", Physical Review A 106 6, 062410 (2022).

[7] Zhian Jia, Minjeong Song, and Dagomir Kaszlikowski, "Quantum space-time marginal problem: global causal structure from local causal information", New Journal of Physics 25 12, 123038 (2023).

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

[9] Shravan Shravan, Simon Morelli, Otfried Gühne, and Satoya Imai, "Geometry of two-body correlations in three-qubit states", arXiv:2309.09549, (2023).

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The above citations are from Crossref's cited-by service (last updated successfully 2024-05-12 09:08:05) and SAO/NASA ADS (last updated successfully 2024-05-12 09:08:06). The list may be incomplete as not all publishers provide suitable and complete citation data.

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