In this paper, we introduce post-selection games, a generalization of nonlocal games where each round can be not only won or lost by the players, but also discarded by the referee. Such games naturally formalize possibilistic proofs of nonlocality, such as Hardy's paradox. We develop algorithms for computing the local and Tsirelson bounds of post-selection games. Furthermore, we show that they have an unbounded advantage in statistical power over traditional nonlocal games, making them ideally suited for analysing Bell tests with low detection efficiency.
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13 pages, 2 figures. v2: added citations and link to repository
In this paper, we introduce post-selection games, a generalization of nonlocal games where each round can be not only won or lost by the players, but also discarded by the referee. Such games naturally formalize possibilistic proofs of nonlocality, such as Hardy's paradox. We develop algorithms for computing the local and Tsirelson bounds of post-selection games. Furthermore, we show that they have an unbounded advantage in statistical power over traditional nonlocal games, making them ideally suited for analysing Bell tests with low detection efficiency.
In order to reject the local hidden variables hypothesis, the usefulness of a Bell inequality can be quantified by how small a p -value it will give for a physical experiment. Here we show that to obtain a small expected p -value it is sufficient to have a large gap between the local and Tsirelson bounds of the Bell inequality, when it is formulated as a nonlocal game. We develop an algorithm for transforming an arbitrary Bell inequality into an equivalent nonlocal game with the largest possible gap, and show its results for the CGLMP and Inn22 inequalities.We present explicit examples of Bell inequalities with gap arbitrarily close to one, and show that this makes it possible to reject local hidden variables with arbitrarily small p -value in a single shot, without needing to collect statistics. We also develop an algorithm for calculating local bounds of general Bell inequalities which is significantly faster than the naïve approach, which may be of independent interest.
Nonlocal games provide a unified framework for studying the distinction between classical, quantum, and more general no-signaling correlations. This review connects 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. We then introduce nonlocal games through the standard predicate/verifier formalism and discuss representative examples, including XOR games, the GHZgame, graph-based colouring games, the Mermin–Peres magic square game, and Hardy’s paradox as a related logical manifestation of nonlocality. The central contribution of the article is a comparative organisation of four closely related representation frameworks: conditional-probability andcorrelation descriptions, Bell-functional formulations, entangled-value optimisation, and the quantum-operator approach together with the Navascu´es–Pironio–Ac´ın hierarchy. These viewpoints are instantiated for the CHSH, magic square, and GHZ games, showing how each representation emphasises a different aspect of the same underlying task. The resulting synthesis clarifies the relation between Bell inequality violations, perfect quantum strategies, pseudo-telepathy, and semidefinite relaxations of quantum correlations.
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.
The notions of entanglement and nonlocality are among the most striking ingredients found in quantum information theory. One tool to better understand these notions is the model of nonlocal games; a mathematical framework that abstractly models a physical system. The simplest instance of a nonlocal game involves two players, Alice and Bob, who are not allowed to communicate with each other once the game has started and who play cooperatively against an adversary referred to as the referee. The focus of this thesis is a class of games called extended nonlocal games, of which nonlocal games are a subset. In an extended nonlocal game, the players initially share a tripartite state with the referee. In such games, the winning conditions for Alice and Bob may depend on outcomes of measurements made by the referee, on its part of the shared quantum state, in addition to Alice and Bob's answers to the questions sent by the referee. We build up the framework for extended nonlocal games and study their properties and how they relate to nonlocal games.
Quantum nonlocality is a counterintuitive phenomenon that lies beyond the purview of causal influences. Recently, Bell inequalities have been generalized to the case of quantum inputs, leading to a powerful family of semiquantum Bell inequalities that are capable of detecting any entangled state. Here, we focus on a different problem and investigate how the local indistinguishability of quantum inputs and postselection may affect the requirements to detect semiquantum nonlocality. To this end, we consider a semiquantum nonlocal game based on locally indistinguishable qubit inputs, and derive its postselected local and quantum bounds by using a connection to the local distinguishability of quantum states. Interestingly, we find that the postselected local bound is independent of the measurement efficiency, and the achievable postselected Bell violation increases with decreasing measurement efficiency.
The predictions of quantum mechanics cannot be resolved with a completely classical view of the world. In particular, the statistics of space-like separated \nmeasurements on entangled quantum systems violate a Bell inequality [Bell1964]. \nWe put forward a computational perspective on a broad class of Bell tests that study correlators, or the statistics of joint measurement outcomes. We associate \nparticular maps, or functions to particular theories. The violation of a Bell \ninequality then implies the ability to perform some functions, or computations \nthat classical, or more generally, local hidden variable (LHV) theories cannot. \nWe derive an in�finite class of Bell inequalities that establish a link to so-called "non-local games" [Cleve2004]. We then make the connection between \nRaussendorf and Briegel's formulation of Measurement-based Quantum Computing (MBQC) [Raussendorf2001], and these non-local games. Not only can we show that a quantum violation implies a computational advantage in this model, \nwe show that adaptive measurements are required to perform all quantum computations. \nFinally, we explore post-selection of data in Bell tests from both a practical and \nconceptual point-of-view, with particular consideration to so-called "loopholes". \nLoopholes allow LHV theories to simulate quantum correlations through post-selection. \nWe give a computational description of how loopholes can emerge in diff�erent post-selection scenarios. This motivates us to �find a form of post-selection \nthat does not lead to loopholes. Central again to this discussion is the \ndescription of LHV theories in terms of computations. \nInterestingly, quantum correlators can be made more "non-classical" with this \nloophole-free post-selection. This method of post-selection also can simulate \ninformation processing tasks, such as MBQC, that have time-like separated components. \nThis opens up new avenues for the study of time-like tasks studied \nwithin the space-like separated scenario of the Bell test.