Bell nonlocality is a cornerstone of quantum theory with applications in information processing ranging from cryptography to distributed computing and game theory. Indeed, it is known that Bell's theorem can be formally linked to Bayesian games, allowing the use of nonlocal correlations to advise players and thereby achieve new points of equilibrium that are unavailable classically. Here we generalize this link, proving the connection between multistage games of incomplete information with Bell scenarios involving the communication of measurement outcomes between the parties. We apply the general framework for a few cases of interest and analyze the equilibrium reached by quantum nonlocal correlations.
We report an experimental demonstration of entanglement swapping over two quantum stages. By successful realizations of two cascaded photonic entanglement swapping processes, entanglement is generated and distributed between two photons, that originate from independent sources and do not share any common past. In the experiment we use three pairs of polarization entangled photons and conduct two Bell-state measurements (BSMs) one between the first and second pair, and one between the second and third pair. This results in projecting the remaining two outgoing photons from pair 1 and 3 into an …
Nonlocality, one of the most remarkable aspects of quantum mechanics, is closely related to Bayesian game theory. Quantum mechanics can offer advantages to some Bayesian games, if the payoff functions are related to Bell inequalities in some way. Most of these Bayesian games that have been discussed are common interest games. Recently the first conflicting interest Bayesian game is proposed in Phys. Rev. Lett. 114, 020401 (2015). In the present paper we present three new conflicting interest Bayesian games where quantum mechanics offers advantages. The first game is linked with Cereceda inequa…
We investigate the relation between Bell inequalities and nonlocal games by presenting a systematic method for their bilateral conversion. In particular, we show that while to any nonlocal game there naturally corresponds a unique Bell inequality, the converse is not true. As an illustration of the method we present a number of nonlocal games that admit better odds when played using quantum resources
I draw attention to statistical, probabilistic, computer science aspects of the highly related topics of the Bell game and of a possible future Quantum Internet.
We show that, for a continuous set of entangled four-partite states, the task of maximizing the payoff in the symmetric-strategy four-player quantum Minority game is equivalent to maximizing the violation of a four-particle Bell inequality with each observer choosing the same set of two dichotomic observables. We conclude the existence of direct correspondences between (i) the payoff rule and Bell inequalities, and (ii) the strategy and the choice of measured observables in evaluating these Bell inequalities. We also show that such a correspondence between Bell polynomials (in a single plane) …
We study the relation between the quantum games, communication complexity problems and Bell inequalities. In particular we are interested in answering the question whether for every element of one of these groups there is a corresponding element in the other two. We show that there are cases where there is no such relation. Moreover, in the communication complexity problems for which there is no Bell inequality the advantage of the quantum strategies over the classical ones is much higher.
The interaction of competing agents is described by classical game theory. It is now well known that this can be extended to the quantum domain, where agents obey the rules of quantum mechanics. This is of emerging interest for exploring quantum foundations, quantum protocols, quantum auctions, quantum cryptography, and the dynamics of quantum cryptocurrency, for example. In this paper, we investigate two-player games in which a strategy pair can exist as a Nash equilibrium when the games obey the rules of quantum mechanics. Using a generalized Einstein-Podolsky-Rosen (EPR) setting for two-pla…
We report constructing quantum games directly from a system of Bell's inequalities using Arthur Fine's analysis published in early 1980s. This analysis showed that such a system of inequalities forms a set of both necessary and sufficient conditions required to find a joint distribution function compatible with a given set of joint probabilities, in terms of which the system of Bell's inequalities is usually expressed. Using the setting of a quantum correlation experiment for playing a quantum game, and considering the examples of Prisoners' Dilemma and Matching Pennies, we argue that this app…
Motivated by the recent success of reinforcement learning in games such as Go and Dota2, we formulate Bell non-local games as a reinforcement learning problem. Such a formulation helps us to explore Bell non-locality in a range of scenarios. The measurement settings and the quantum states are selected by the learner randomly in the beginning. Still, eventually, it starts understanding the underlying patterns and discovers an optimal (or near-optimal) quantum configuration corresponding to the task at hand. We provide a proof of principle approach to learning quantum configurations for violatin…
We discuss a connection between Bell nonlocality and Bayesian games. This link offers interesting perspectives for Bayesian games, namely to allow the players to receive advice in the form of nonlocal correlations, for instance using entangled quantum particles or more general no-signaling boxes. The possibility of having such 'nonlocal advice' will lead to novel joint strategies, impossible to achieve in the classical setting. This implies that quantum resources, or more general no-signaling resources, offer a genuine advantage over classical ones. Moreover, some of these strategies can repre…
In a recent paper, Junge and Palazuelos presented two two-player games exhibiting interesting properties. In their first game, entangled players can perform notably better than classical players. The quantitative gap between the two cases is remarkably large, especially as a function of the number of inputs to the players. In their second game, entangled players can perform notably better than players that are restricted to using a maximally entangled state (of arbitrary dimension). This was the first game exhibiting such a behavior. The analysis of both games is heavily based on non-trivial r…
Nonlocal games with synchronous correlations are a natural generalization of functions between two finite sets. In this work we examine analogues of Bell's inequalities for such correlations, and derive a synchronous device-independent quantum key distribution protocol. This protocol has the advantage of symmetry between the two users and self-testing while generating shared secret key without requiring a preshared secret. We show that, unlike general correlations and the CHSH inequality, there can be no quantum Bell violation among synchronous correlations with two measurement settings. Howev…
Quantum state targeting is a quantum game which results from combining traditional quantum state estimation with additional classical information. We consider a particular version of the game and show how it can be played with maximally entangled states. The optimal solution of the game is used to derive a Bell inequality for two entangled qutrits. We argue that the nice properties of the inequality are direct consequences of the method of construction.
Games with incomplete information are formulated in a multi-sector probability matrix formalism that can cope with quantum as well as classical strategies. An analysis of classical and quantum strategy in a multi-sector extension of the game of Battle of Sexes clarifies the two distinct roles of nonlocal strategies, and establish the direct link between the true quantum gain of game's payoff and the breaking of Bell inequalities.
Recently, Fan \textit{et al.} [Mod. Phys. Lett. A 36, 2150223 (2021)], presented a generalized Clauser-Horne-Shimony-Holt (CHSH) inequality, to identify $N$-qubit Greenberger-Horne-Zeilinger (GHZ) states. They showed an interesting phenomenon that the maximal violation of the generalized CHSH inequality is robust under some specific noises. In this work, we map the inequality to the CHSH game, and consequently to the CHSH* game in a single-qubit system. This mapping provides an explanation for the robust violations in $N$-qubit systems. Namely, the robust violations, resulting from the degener…
We study the classical and quantum values of one- and two-party linear games, an important class of unique games that generalizes the well-known XOR games to the case of non-binary outcomes. We introduce a ``constraint graph" associated to such a game, with the constraints defining the linear game represented by an edge-coloring of the graph. We use the graph-theoretic characterization to relate the task of finding equivalent games to the notion of signed graphs and switching equivalence from graph theory. We relate the problem of computing the classical value of single-party anti-correlation …
Nonlocal game as a novel witness of the nonlocality of entanglement is of fundamental importance in various fields. The known nonlocal games or equivalent linear Bell inequalities are only useful for Bell networks of single entanglement. Our goal in this paper is to propose a unified method for constructing cooperating games in network scenarios. We first propose an efficient method to construct numerous multipartite games from any graphs. The main idea is the graph representation of entanglement-based quantum networks. We further specify these graphic games with quantum advantages by providin…
Pseudo-telepathy is the most recent form of rejection of locality. Many of its properties have already been discovered: for instance, the minimal entanglement, as well as the minimal cardinality of the output sets, have been characterized. This paper contains two main results. First, we prove that no bipartite pseudo-telepathy game exists, in which one of the partners receives only two questions; as a corollary, we show that the minimal "input cardinality", that is, the minimal number of questions required in a bipartite pseudo-telepathy game, is 3x3. Second, we study the Bell-type inequality …
We present a game-based approach to teach Bell inequalities and quantum cryptography at high school. The approach is based on kinesthetic activities and allows students to experience and discover quantum features and their applications first-hand. We represent quantum states by the orientation of students, and mimic quantitative random behaviour and measurements using dice and apps.
Entangled quantum systems can exhibit correlations that cannot be simulated classically. For historical reasons such correlations are called "Bell inequality violations." We give two new two-player games with Bell inequality violations that are stronger, fully explicit, and arguably simpler than earlier work. The first game is based on the Hidden Matching problem of quantum communication complexity, introduced by Bar-Yossef, Jayram, and Kerenidis. This game can be won with probability 1 by a strategy using a maximally entangled state with local dimension $n$ (e.g., $\log n$ EPR-pairs), while w…
Categorical quantum mechanics, which examines quantum theory via dagger-compact closed categories, gives satisfying high-level explanations to the quantum information procedures such as Bell-type entanglement or complementary observables (\cite{AC}, \cite{Co}, \cite{Co2}). Inspired by the fact that Quantum Game Theory can be seen as branch of quantum information, we express Quantum Game Theory procedures using the topological semantics provided by Categorical Quantum Mechanics. We also investigate Bayesian Games with correlation from this novel point of view while considering the connection be…
We provide an interesting two-party parity oblivious communication game whose success probability is solely determined by the Bell expression. The parity-oblivious condition in an operational quantum theory implies the preparation non-contextuality in an ontological model of it. We find that the aforementioned Bell expression has two upper bounds in an ontological model; the usual local bound and a non-trivial preparation non-contextual bound arising from the non-trivial parity-oblivious condition, which is smaller that the local bound. We first demonstrate the communication game when both Ali…
According to recent reports, the last loopholes in testing Bell's inequality are closed. It is argued that the really important task in this field has not been tackled yet and that the leading experiments claiming to close locality and detection efficiency loopholes, although making a very significant progress, have conceptual drawbacks. The important task is constructing quantum devices which will allow winning games of certain correlated replies against any classical team. A novel game of this type is proposed.
John Bell taught us that quantum mechanics can not be reproduced by non-contextual and local Hidden variable theory. The impossibility of replacing quantum mechanics by non-contextual Hidden Variable Theory can be turned to a impossible coloring pseudo-telepathy game to be played by two distant players. The game can not be won without communication in the classical world. But if the players share entangled state (quantum correlation) the game can be won deterministically using no communication. This again shows that though quantum correlation can not be used for communication, two parties can …
Certification of quantum systems and operations is a central task in quantum information processing. Most current schemes rely on a tomography with fully characterised devices, while this may not be met in real experiments. Device characterisations can be removed with device-independent tests, it is technically challenging at the moment, though. In this letter, we investigate the problem of certifying entangled states and measurements via semi-quantum games, a type of non-local quantum games with well characterised quantum inputs, balancing practicality and device-independence. We first design…
The Bell experiment is a random game with two binary outcomes whose statistical correlation is given by $E_0(Θ)=-\cos(Θ)$, where $Θ\in [-π, π)$ is an angular input that parameterizes the game setting. The correlation function $E_0(Θ)$ belongs to the affine space ${\cal H} \equiv \left\{E(Θ)\right\}$ of all continuous and differentiable periodic functions $E(Θ)$ that obey the parity symmetry constraints $E(-Θ)=E(Θ)$ and $E(π-Θ)=-E(Θ)$ with $E(0)=-1$ and, furthermore, are strictly monotonically increasing in the interval $[0, π)$. Here we show how to build explicitly local statistical models of …
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 whether every proper (tight) 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…
The phenomenon of monogamy of Bell inequality violations is interesting both from the fundamental perspective as well as in cryptographic applications such as the extraction of randomness and secret bits. In this article, we derive new and stronger monogamy relations for violations of Bell inequalities in general no-signaling theories. These relations are applicable to the class of binary output correlation inequalities known as XOR games, and to a restricted set of unique games. In many instances of interest, we show that the derived relation provides a significant strengthening over previous…
We analyze the difference between ex ante and ex post equilibria in classical games played with the assistance of a nonlocal (quantum or no-signaling) resource. In physics, the playing of these games is known as performing bipartite Bell-type experiments. By analyzing the Clauser-Horn-Shimony-Holt game, we find a constructive procedure to find two-person Bayesian games with a nonlocal (i.e. no-signaling, and, in many cases, quantum) advantage. Most games of this kind known from the literature can be constructed along this principle, and share the property that their relevant ex ante equilibria…
In this work we introduce an intermediate setting between quantum nonlocality and communication complexity problems. More precisely, we study the value of XOR games $G$ when Alice and Bob are allowed to use a limited amount of one-way classical communication $ω_{o.w.-c}(G)$ (resp. one-way quantum communication $ω_{o.w.-c}^*(G)$), where $c$ denotes the number of bits (resp. qubits). The key quantity here is the quotient $ω_{o.w.-c}^*(G)/ω_{o.w.-c}(G)$. We provide a universal way to obtain Bell inequality violations of general Bell functionals from XOR games for which the quotient $ω_{o.w.-c}^*(…
The boundary between classical and quantum correlations is well characterised by linear constraints called Bell inequalities. It is much harder to characterise the boundary of the quantum set itself in the space of no-signaling correlations. For the points on the quantum boundary that violate maximally some Bell inequalities, Oppenheim and Wehner [Science 330, 1072 (2010)] pointed out a complex property: the optimal measurements of Alice steer Bob's local state to the eigenstate of an effective operator corresponding to its maximal eigenvalue. This effective operator is the linear combination …
Self-testing is a method to verify that one has a particular quantum state from purely classical statistics. For practical applications, such as device-independent delegated verifiable quantum computation, it is crucial that one self-tests multiple Bell states in parallel while keeping the quantum capabilities required of one side to a minimum. In this work, we use the $3 \times n$ magic rectangle games (generalizations of the magic square game) to obtain a self-test for $n$ Bell states where the one side needs only to measure single-qubit Pauli observables. The protocol requires small input s…
We present a consistent formulation of quantum game theory that accommodates all possible strategies in Hilbert space. The physical content of the quantum strategy is revealed as a family of classical games representing altruistic game play supplemented by quantum interferences. Crucial role of the entanglement in quantum strategy is illustrated by an example of quantum game representing the Bell's experiment.
The explicit construction is presented of two-player game satisfying: (i) symmetry with respect to the permutation of the players; (ii) the existence of upper bound on total payoff following from Bell inequality; (iii) the existence of unfair equilibrium with total payoff saturating the above bound. The quantum counterpart of the game is considered which possesses only fair equilibria and strategies outperforming the classical ones.
We propose a scheme for a quantum game based on performing an EPR type experiment and in which each player's spatial directional choices are considered as their strategies. A classical mixed-strategy game is recovered by restricting the players' choices to specific spatial trajectories. We show that for players' directional choices for which the Bell-CHSH inequality is violated, the players' payoffs in the quantum game have no mapping within the classical mixed-strategy game. The scheme provides a more direct link between classical and quantum games.
We propose a set of Bell-type nonlocal games that can be used to prove an unconditional quantum advantage in an objective and hardware-agnostic manner. In these games, the circuit depth needed to prepare a cyclic cluster state and measure a subset of its Pauli stabilizers on a quantum computer is compared to that of classical Boolean circuits with the same, nearest-neighboring gate connectivity. Using a circuit-based trapped-ion quantum computer, we prepare and measure a six-qubit cyclic cluster state with an overall fidelity of 60.6% and 66.4%, before and after correcting for measurement-read…
Research in quantum games has flourished during recent years. However, it seems that opinion remains divided about their true quantum character and content. For example, one argument says that quantum games are nothing but 'disguised' classical games and that to quantize a game is equivalent to replacing the original game by a different classical game. The present thesis contributes towards the ongoing debate about quantum nature of quantum games by developing two approaches addressing the related issues. Both approaches take Einstein-Podolsky-Rosen (EPR)-type experiments as the underlying phy…
Playing a symmetric bi-matrix game is usually physically implemented by sharing pairs of 'objects' between two players. A new setting is proposed that explicitly shows effects of quantum correlations between the pairs on the structure of payoff relations and the 'solutions' of the game. The setting allows a re-expression of the game such that the players play the classical game when their moves are performed on pairs of objects having correlations that satisfy the Bell's inequalities. If players receive pairs having quantum correlations the resulting game cannot be considered another classical…
A quantum version of the Matching Pennies (MP) game is proposed that is played using an Einstein-Podolsky-Rosen-Bohm (EPR-Bohm) setting. We construct the quantum game without using the state vectors, while considering only the quantum mechanical joint probabilities relevant to the EPR-Bohm setting. We embed the classical game within the quantum game such that the classical MP game results when the quantum mechanical joint probabilities become factorizable. We report new Nash equilibria in the quantum MP game that emerge when the quantum mechanical joint probabilities maximally violate the Clau…
It is by now well-established that there exist non-local games for which the best entanglement-assisted performance is not better than the best classical performance. Here we show in contrast that any two-player XOR game, for which the corresponding Bell inequality is tight, has a quantum advantage. In geometric terms, this means that any correlation Bell inequality for which the classical and quantum maximum values coincide, does not define a facet, i.e. a face of maximum dimension, of the local Bell polytope. Indeed, using semidefinite programming duality, we prove upper bounds on the dimens…
The characterization of the set of quantum correlations is a problem of fundamental importance in quantum information. The question whether every proper (tight) Bell inequality is violated in Quantum theory is an intriguing one in this regard. Here, we make significant progress in answering this question, by showing that every tight Bell inequality is violated by 'Almost Quantum' correlations, a semi-definite programming relaxation of the set of quantum correlations. As a consequence, we show that many (classes of) Bell inequalities including two-party correlation Bell inequalities and multi-o…
Communication games are crucial tools for investigating the limitations of physical theories. The communication complexity (CC) problem is a typical example, for which several distributed parties attempt to jointly calculate a given function with limited classical communications. In this work, we present a method to construct CC problems from Bell tests in a graph-theoretic way. Starting from an experimental compatibility graph and the corresponding Bell test function, a target function which encodes the information of each edge can be constructed, then using this target function we could cons…
