Naturally restricted subsets of nonsignaling correlations: typicality
and convergence
- URL: http://arxiv.org/abs/2107.05646v4
- Date: Tue, 12 Jul 2022 13:28:31 GMT
- Title: Naturally restricted subsets of nonsignaling correlations: typicality
and convergence
- Authors: Pei-Sheng Lin, Tam\'as V\'ertesi, Yeong-Cherng Liang
- Abstract summary: In a Bell experiment, the observed correlation between measurement outcomes can be stronger than that allowed by local causality.
We consider several bipartite Bell scenarios and numerically estimate their volume relative to that of the set of nonsignaling correlations.
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: 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 $n_s$
(outputs $n_o$), the relative volume of both the Bell-local set and the quantum
set increases (decreases) rapidly with increasing $n_o$ ($n_s$) (2) although
the so-called macroscopically local set $Q_1$ may approximate $Q$ well in the
two-input scenarios, it can be a very poor approximation of the quantum set
when $n_s>n_o$ (3) the almost-quantum set $\tilde{Q}_1$ is 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
$n_o=2$, whereas (5) the difference between the Bell-local set and the PPT set
generally becomes more significant with increasing $n_o$. 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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