1Department of Physics and Center for Quantum Frontiers of Research & Technology (QFort), National Cheng Kung University, Tainan 701, Taiwan
2MTA Atomki Lendület Quantum Correlations Research Group, Institute for Nuclear Research, P.O. Box 51, H-4001 Debrecen, Hungary
3Physics Division, National Center for Theoretical Sciences, Taipei 10617, Taiwan
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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$$gt$$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.
Featured image: Illustration of the various naturally restricted subsets of the nonsignaling set $mathcal{NS}$ of correlations and their inclusion relations. In solid lines and moving inwards, we have, respectively, the boundary of the quantum set of correlations $mathcal{Q}$ (blue), the convex hull $mathcal{M}^text{P}$ (red) of the set of correlations attainable by finite-dimensional maximally entangled states in conjunction with projective measurements, the set of correlations attainable by positive-partial-transposed states $mathcal{P}$ (green), and the Bell-local polytope $mathcal{L}$ (sky-blue). The dashed (pink) and dotted (brown) lines that lie between the boundary of $mathcal{Q}$ and that of $mathcal{NS}$ mark the border of the lowest-level outer approximation of $mathcal{Q}$, respectively, due to Navascués-Pironio-Acín ($mathcal{Q}_1$) and Moroder-Bancal-Liang-Hofmann-Gühne ($tilde{mathcal{Q}}_1)$. Accordingly, these are also known as the macroscopically-local set and the almost-quantum set of correlations.
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