TY - JOUR

T1 - Naturally restricted subsets of nonsignaling correlations

T2 - Typicality and convergence

AU - Lin, Pei Sheng

AU - Vértesi, Tamás

AU - Liang, Yeong Cherng

N1 - Funding Information:
We are grateful to Tulja Varun Kondra for his contribution at the initial stage of this research project, to Antonio Acín, Jean-Daniel Bancal, Elie Wolfe, and Denis Rosset for their suggestions on how uniform sampling on convex sets (of correlations) may be performed. Part of this work was completed during PSL’s visit to the Institute of Photonic Sciences (ICFO), Spain and YCL’s visit to the Perimeter Institute for Theoretical Physics, Canada. The hospitality of both institutes is greatly appreciated. This work is supported by the Ministry of Science and Technology, Taiwan (Grants No. 104-2112-M-006-021-MY3, 107-2112-M-006-005-MY2, and 109-2112-M006-010-MY3), the EU (QuantERA eDICT) and the National Research, Development and Innovation Office NKFIH (No. 2019-2.1.7-ERA-NET-2020-00003).
Publisher Copyright:
© 2021 Elsevier B.V.. All rights reserved.

PY - 2022

Y1 - 2022

N2 - It is well-known that in a Bell experiment, the observed correlation between measurement outcomes - as predicted by quantum theory - can be stronger than that allowed by local causality, yet not fully constrained by the principle of relativistic causality. In practice, the characterization of the set Q of quantum correlations is carried out, often, through a converging hierarchy of outer approximations. On the other hand, some subsets of Q arising from additional constraints [e.g., originating from quantum states having positive-partial-transposition (PPT) or being finite-dimensional maximally entangled (MES)] turn out to be also amenable to similar numerical characterizations. How, then, at a quantitative level, are all these naturally restricted subsets of nonsignaling correlations different? Here, we consider several bipartite Bell scenarios and numerically estimate their volume relative to that of the set of nonsignaling correlations. Within the number of cases investigated, we have observed that (1) for a given number of inputs ns(outputs no), the relative volume of both the Bell-local set and the quantum set increases (decreases) rapidly with increasing no (ns) (2) although the so-called macroscopically local set Q1 may approximate Q well in the two-input scenarios, it can be a very poor approximation of the quantum set when ns> no(3) the almost-quantum set Q1is an exceptionally-good approximation to the quantum set (4) the difference between Q and the set of correlations originating from MES is most significant when no= 2, whereas (5) the difference between the Bell-local set and the PPT set generally becomes more significant with increasing no. This last comparison, in particular, allows us to identify Bell scenarios where there is little hope of realizing the Bell violation by PPT states and those that deserve further exploration.

AB - It is well-known that in a Bell experiment, the observed correlation between measurement outcomes - as predicted by quantum theory - can be stronger than that allowed by local causality, yet not fully constrained by the principle of relativistic causality. In practice, the characterization of the set Q of quantum correlations is carried out, often, through a converging hierarchy of outer approximations. On the other hand, some subsets of Q arising from additional constraints [e.g., originating from quantum states having positive-partial-transposition (PPT) or being finite-dimensional maximally entangled (MES)] turn out to be also amenable to similar numerical characterizations. How, then, at a quantitative level, are all these naturally restricted subsets of nonsignaling correlations different? Here, we consider several bipartite Bell scenarios and numerically estimate their volume relative to that of the set of nonsignaling correlations. Within the number of cases investigated, we have observed that (1) for a given number of inputs ns(outputs no), the relative volume of both the Bell-local set and the quantum set increases (decreases) rapidly with increasing no (ns) (2) although the so-called macroscopically local set Q1 may approximate Q well in the two-input scenarios, it can be a very poor approximation of the quantum set when ns> no(3) the almost-quantum set Q1is an exceptionally-good approximation to the quantum set (4) the difference between Q and the set of correlations originating from MES is most significant when no= 2, whereas (5) the difference between the Bell-local set and the PPT set generally becomes more significant with increasing no. This last comparison, in particular, allows us to identify Bell scenarios where there is little hope of realizing the Bell violation by PPT states and those that deserve further exploration.

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U2 - 10.22331/Q-2022-07-14-765

DO - 10.22331/Q-2022-07-14-765

M3 - Article

AN - SCOPUS:85136157925

SN - 2521-327X

VL - 6

JO - Quantum

JF - Quantum

ER -