Naturally restricted subsets of nonsignaling correlations: typicality and convergence

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.

v4: published version (in Quantum); v3: Substantially rewritten, main results summarized in 10 observations, 8 figures, and 7 tables; v2: Results updated; v1: 13 + 4 pages, 10 tables, 5 figures, this is [66] of arXiv:1810.00443; Comments are welcome!

References in corpus (9)

• Device-independent security of quantum cryptography against collective attacks
• A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations
• Bounding the set of quantum correlations
• Almost quantum correlations
• Disproving the Peres conjecture: Bell nonlocality from bipartite bound entanglement
• Device-independent certification of entangled measurements
• Semi-device-independent bounds on entanglement
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Cited by in corpus (7)

• Semidefinite programming relaxations for quantum correlations
• Semi-definite programming and quantum information
• Quantum correlations on the no-signaling boundary: self-testing and more
• Optimal tests of genuine multipartite nonlocality
• Semi-device-independently characterizing quantum temporal correlations
• Bell inequality violations with random mutually unbiased bases
• Device-independent certification of desirable properties with a confidence interval