A hierarchy of semidefinite programs for generalised Einstein-Podolsky-Rosen scenarios

Matty J. Hoban; Tom Drescher; Ana Belén Sainz

doi:10.22331/q-2025-01-14-1591

A hierarchy of semidefinite programs for generalised Einstein-Podolsky-Rosen scenarios

Matty J. Hoban¹, Tom Drescher², and Ana Belén Sainz^3,4

¹Quantum Group, Department of Computer Science, University of Oxford, United Kingdom
²University of Innsbruck, Department of Mathematics, A-6020 Innsbruck, Austria
³International Centre for Theory of Quantum Technologies, University of Gdańsk, 80-309 Gdańsk, Poland
⁴Basic Research Community for Physics e.V., Germany

Published:	2025-01-14, volume 9, page 1591
Editor:	Ujjwal Sen
Eprint:	arXiv:2208.09236v2
Doi:	https://doi.org/10.22331/q-2025-01-14-1591
Citation:	Quantum 9, 1591 (2025).

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Abstract

Correlations in Einstein-Podolsky-Rosen (EPR) scenarios, captured by $assemblages$ of unnormalised quantum states, have recently caught the attention of the community, both from a foundational and an information-theoretic perspective. The set of quantum-realisable assemblages, or abbreviated to quantum assemblages, are those that arise from multiple parties performing local measurements on a shared quantum system. In general, deciding whether or not a given assemblage is a quantum assemblage, i.e. membership of the set of quantum assemblages, is a hard problem, and not always solvable. In this paper we introduce a hierarchy of tests where each level either determines non-membership of the set of quantum assemblages or is inconclusive. The higher the level of the hierarchy the better one can determine non-membership, and this hierarchy converges to a particular set of assemblages. Furthermore, this set to which it converges contains the quantum assemblages. Each test in the hierarchy is formulated as a semidefinite program. This hierarchy allows one to upper bound the quantum violation of a steering inequality and the quantum advantage provided by quantum EPR assemblages in a communication or information-processing task.

Popular summary

Quantum entanglement enables different forms of correlation between measurements made on multiple physical systems. One such form is that of Einstein-Podolsky-Rosen (EPR) "steering", which can trace its origins back to the 1935 seminal work of these authors; this phenomenon has been re-examined since the birth of quantum information and shown to be a resource in various information processing tasks. To understand this resource it is essential to characterise the amount of the resource present in these EPR experiments. Our work gives a general framework for how to bound the amount of "steering" present in a resource, and in a setting that can go beyond conventional experiments. This hopefully opens up new possibilities for understanding what is special about quantum theory and if there are new information processing tasks that can exploit this "generalised" resource.

► BibTeX data

@article{Hoban2025hierarchyof,
  doi = {10.22331/q-2025-01-14-1591},
  url = {https://doi.org/10.22331/q-2025-01-14-1591},
  title = {A hierarchy of semidefinite programs for generalised {E}instein-{P}odolsky-{R}osen scenarios},
  author = {Hoban, Matty J. and Drescher, Tom and Sainz, Ana Bel{\'{e}}n},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1591},
  month = jan,
  year = {2025}
}

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Cited by

[1] Beata Zjawin, Matty J. Hoban, Paul Skrzypczyk, and Ana Belén Sainz, "Activation of postquantumness in bipartite generalized Einstein-Podolsky-Rosen scenarios", Physical Review A 110 4, 042212 (2024).

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