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arXiv:2310.10600 · Physical Review Research

Equivalence between face nonsignaling correlations, full nonlocality, all-versus-nothing proofs, and pseudotelepathy

Yuan Liu, Ho Yiu Chung, Emmanuel Zambrini Cruzeiro, Junior R. Gonzales-Ureta, Ravishankar Ramanathan, Adán Cabello

Published Nov 4, 2024• 11 citations
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Abstract

We show that a quantum correlation p is in a face of the nonsignaling polytope with no local points if and only if p has nonlocal content 1, if and only if p allows for a Greenberger-Horne-Zeilinger-like proof, and if and only if p provides a perfect strategy for a nonlocal game. That is, face nonsignaling (FNS) correlations, full nonlocality (FN), all-versus-nothing (AVN) proofs, and pseudotelepathy (PT) are equivalent. This shows that different resources behind a wide variety of fundamental results are in fact the same resource. We demonstrate that quantum correlations with FNS = FN = AVN = PT do not need to maximally violate a tight Bell inequality. We introduce a method for identifying quantum FNS = FN = AVN = PT correlations and use it to prove quantum mechanics does not allow for FNS = FN = AVN = PT neither in the ( 3 , 3 ; 3 , 2 ) nor in the ( 3 , 2 ; 3 , 4 ) Bell scenarios. This solves an open problem that, because of the FNS = FN = AVN = PT equivalence, has implications in several fields. Published by the American Physical Society 2024

DOI
10.1103/physrevresearch.6.l042035

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We show that a quantum correlation p is in a face of the nonsignaling polytope with no local points if and only if p has nonlocal content 1, if and only if p allows for a Greenberger-Horne-Zeilinger-like proof, and if and only if p provides a perfect strategy for a nonlocal game. That is, face nonsignaling (FNS) correlations, full nonlocality (FN), all-versus-nothing (AVN) proofs, and pseudotelepathy (PT) are equivalent. This shows that different resources behind a wide variety of fundamental results are in fact the same resource. We demonstrate that quantum correlations with FNS = FN = AVN = PT do not need to maximally violate a tight Bell inequality. We introduce a method for identifying quantum FNS = FN = AVN = PT correlations and use it to prove quantum mechanics does not allow for FNS = FN = AVN = PT neither in the ( 3 , 3 ; 3 , 2 ) nor in the ( 3 , 2 ; 3 , 4 ) Bell scenarios. This solves an open problem that, because of the FNS = FN = AVN = PT equivalence, has implications in several fields. Published by the American Physical Society 2024
It is a well-established fact that some quantum correlations can be nonlocal, meaning that they cannot be described by a local hidden variable model. Certain quantum correlations have a form of nonlocality so strong that they cannot be reproduced even by models having an arbitrarily small local hidden variable component. These correlations are called fully nonlocal and lead to Bell inequalities in which the maximum quantum value saturates the non-signaling bound. A well-known example of this effect, which is also referred to as quantum pseudo-telepathy or all-versus-nothing proofs of nonlocality, is the quantum distribution fulfilling the Peres-Mermin square, in which the underlying state is a 4×44\times44×4 dimensional maximally entangled state. Other examples of full nonlocality are known but, so far, all of them are for maximally entangled states and it is an open question whether maximal entanglement is necessary for full nonlocality. In this work, we first establish a link between full nonlocality and the concept of antidistinguishability of quantum states. We use this connection to show that in every bipartite d×dd\times dd×d Hilbert space, with d≥3d\geq3d≥3, there are non-maximally entangled states that are fully nonlocal. In fact, we derive simple sufficient conditions for full nonlocality that are only based on the smallest and largest Schmidt coefficients. We also show that in every dimension there exist pure entangled states that do not exhibit full nonlocality. Finally, we show that all pure entangled states can be activated to show full nonlocality in the many-copy scenario.
We study the faces of the set of quantum correlations, i.e., the Bell and noncontextuality inequalities without any quantum violation. First, we investigate the question of whether every proper (facet-defining) Bell inequality for two parties, other than the trivial ones from positivity, normalization, and no-signaling, can be violated by quantum correlations, i.e., whether the classical Bell polytope or the smaller correlation polytope share any facets with their respective quantum sets. To do this, we develop a recently derived bound on the quantum value of linear games based on the norms of game matrices to give a simple sufficient condition to identify linear games with no quantum advantage. Additionally we show how this bound can be extended to the general class of unique games. We then show that the paradigmatic examples of correlation Bell inequalities with no quantum violation, namely the nonlocal computation games, do not constitute facet-defining Bell inequalities, not even for the correlation polytope. We also extend this to an arbitrary prime number of outcomes for a specific class of these games. We then study the faces in the simplest Clauser-Horne-Shimony-Holt Bell scenario of binary dichotomic measurements, and identify edges in the set of quantum correlations in this scenario. Finally, we relate the noncontextual polytope of single-party correlation inequalities with the cut polytope $\text{CUT}(\mathbf{\ensuremath{\nabla}}G)$, where GGG denotes the compatibility graph of observables in the contextuality scenario and $\mathbf{\ensuremath{\nabla}}G$ denotes the suspension graph of GGG. We observe that there exist facet-defining noncontextuality inequalities with no quantum violation, and furthermore that this set of inequalities is beyond those implied by the consistent exclusivity principle.
Bell nonlocality places the long-standing philosophical disputes over the nature of quantum theory on a mathematically rigorous, experimentally testable footing, positioning quantum mechanics between local realism and the most general no-signaling theories. Yet the precise quantum boundary has remained elusive even in the minimal Bell scenario, where the extreme points generating the set are known. Here we characterize this boundary through a computable necessary and sufficient criterion: a correlation is quantum if and only if it can be decomposed into an extreme quantum correlation and a classical correlation. The criterion follows from our proof that every Bell functional exhibiting quantum advantage admits a unique quantum maximizer. We further classify the finitely many maximal-face structures containing each nonlocal exposed point according to their local deterministic vertices. By sharply distinguishing physically realizable correlations from nonphysical ones, these results resolve a long-standing open problem and provide a new foundation for identifying the principles underlying quantum theory and determining the ultimate information-theoretic limits of quantum correlations.
Quantum mechanics is a nonlocal theory, but not as nonlocal as the no-signalling principle allows. However, there exist quantum correlations that exhibit maximal nonlocality: they are as nonlocal as any nonsignalling correlation and thus have a local content, quantified by the fraction pL{p}_{L}pL​ of events admitting a local description, equal to zero. We exploit the known link between the Kochen-Specker and Bell theorems to derive a maximal violation of a Bell inequality from every Kochen-Specker proof. We then show that these Bell inequalities lead to experimental bounds on the local content of quantum correlations that are significantly better than those based on other constructions. We perform the experimental demonstration of a Bell test originating from the Peres-Mermin Kochen-Specker proof, providing an upper bound on the local content ${p}_{L}\ensuremath{\lesssim}0.22$.
Nonlocal games provide a unified framework for studying the distinction between classical, quantum, and more general no-signaling correlations. In this work, we develop this perspective by connecting the Bell-locality framework to several complementary mathematical representations of nonlocal games and quantum strategies. We begin with local hidden-variable models, the CHSH inequality, and the role of Bell nonlocality as a device-independent witness of entanglement, and then introduce nonlocal games through the standard predicate/verifier formalism. We next examine a set of representative examples, including XOR games, the GHZ game, graph-based coloring games, the Mermin-Peres magic square game, and Hardy's paradox as a related logical manifestation of nonlocality. Building on this foundation, we compare four closely related representation frameworks: conditional-probability and correlation descriptions, Bell-functional formulations, entangled-value optimization, and the quantum-operator approach together with the Navascues-Pironio-Acin (NPA) hierarchy. These viewpoints are then instantiated for the CHSH, magic square, and GHZ games, showing how each representation emphasizes a different aspect of the same underlying task. Taken together, these examples show that nonlocal games can be studied simultaneously as geometric objects in correlation space, optimization problems over entangled resources, and operator-theoretic constructions. This multi-representation viewpoint clarifies the relation between Bell inequality violations, perfect quantum strategies, pseudo-telepathy, and semidefinite relaxations of quantum correlations.