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Quantum Information and Computation

Can quantum entanglement implement classical correlated equilibria

DeckelbaumAlan

Published Apr 1, 2014• 0 citations
Publisher

Abstract

We ask whether players of a classical game can partition a pure quantum state to implement classical correlated equilibrium distributions. The main contribution of this work is an impossibility res...
DOI
10.5555/2638661.2638668

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We ask whether players of a classical game can partition a pure quantum state to implement classical correlated equilibrium distributions. The main contribution of this work is an impossibility result: we provide an example of a classical correlated equilibrium that cannot be securely implemented without useful information leaking outside the system. We study the model where players of a classical complete information game initially share an entangled pure quantum state. Players may perform arbitrary local operations on their subsystems, but no direct communication (either quantum or classical) is allowed. We explain why, for the purpose of implementing classical correlated equilibria, it is desirable to restrict the initial state to be pure and to restrict communication. In this framework, we define the concept of pure quantum correlated equilibrium (PQCE) and show that in a normal form game, any outcome distribution implementable by a PQCE can also be implemented by a classical correlated equilibrium (CE), but that the converse is false. We extend our analysis to extensive form games, and compare the power of PQCE to extensive form classical correlated equilibria (EFCE) and immediate-revelation extensive form correlated equilibria (IR-EFCE).
We study the scenario where the players of a classical complete information game initially share an entangled pure quantum state. Each player may perform arbitrary local operations on his own qubits, but no direct communication is allowed. In this framework, we define the concept of quantum correlated equilibrium (QCE) for both normal and extensive form games of complete information. We show that in a normal form game, any outcome distribution implementable by a QCE can also be implemented by a classical correlated equilibrium (CE). We prove that the converse is surprisingly false: we give an example of an outcome distribution of a normal form game which is implementably by a CE, yet we prove that in any attempted quantum protocol beginning with a partition of a pure quantum state, at least one of the players will have incentive to deviate. We extend our analysis to extensive form games, and find that the relation between classical and quantum correlated equilibria becomes less clear. We compare the outcome distributions implementable in our quantum model to those implementable by a classical extensive form correlated equilibrium (EFCE). For example, we show that there exists an extensive form complete information game and a distribution of outcomes which can be implemented by a QCE but not by any EFCE, in contrast to the result for normal form games. We also consider the concept of an immediate-revelation extensive form correlated equilibrium (IR-EFCE) and compare the power of IR-EFCE to EFCE and to QCE.
Quantum game theory is a multidisciplinary field which combines quantum mechanics with game theory by introducing non-classical resources such as entanglement, quantum operations and quantum measurement. By transferring two-player two-strategy (2 × 2) dilemma containing classical games into the quantum realm, dilemmas can be resolved in quantum pure strategies if entanglement is distributed between the players who use quantum operations. Moreover, players receive the highest sum of payoffs available in the game, which are otherwise impossible in classical pure strategies. Encouraged by the observation of rich dynamics of physical systems with many interacting parties and the power of entanglement in quantum versions of 2 × 2 games, it became generally accepted that quantum versions can be easily extended to N -player situations by simply allowing N -partite entangled states. In this article, however, we show that this is not generally true because the reproducibility of classical tasks in the quantum domain imposes limitations on the type of entanglement and quantum operators. We propose a benchmark for the evaluation of quantum and classical versions of games, and derive the necessary and sufficient conditions for a physical realization. We give examples of entangled states that can and cannot be used, and the characteristics of quantum operators used as strategies.
A number of recent studies have focused on novel features in game theory when the games are played using quantum mechanical toolbox (entanglement, unitary operators, measurement). Researchers have concentrated in two-player-two strategy, 2x2, dilemma containing classical games, and transferred them into quantum realm showing that in quantum pure strategies dilemmas in such games can be resolved if entanglement is distributed between the players armed with quantum operations. Moreover, it became clear that the players receive the highest sum of payoffs available in the game, which are otherwise impossible in classical pure strategies. Encouraged by the observation of rich dynamics of physical systems with many interacting parties and the power of entanglement in quantum versions of 2x2 games, it became generally accepted that quantum versions can be easily extended to N-player situations by simply allowing N-partite entangled states. In this article, however, we show that this is not generally true because the reproducibility of classical tasks in quantum domain imposes limitations on the type of entanglement and quantum operators. We propose a benchmark for the evaluation of quantum and classical versions of games, and derive the necessary and sufficient conditions for a physical realization. We give examples of entangled states that can and cannot be used, and the characteristics of quantum operators used as strategies.
Bayesian games, also known as games of incomplete information, are a fruitful arena for exploring the impact of correlations on a set of independent agents (players) via the game equilibria to which they give rise. It was realised some time ago that quantum states shared between the players can lead to new and beneficial equilibria, compared to classical correlation. While until now examples of this effect required an entangled state, here we show that even separable states can create new, genuinely quantum equilibria in games, that are advantageous with respect to all classically correlated equilibria. This shows that non-classical correlations beyond entanglement are indeed a resource, even in otherwise entirely classical situations. Our result brings quantum advantage in games significantly closer to possible realisation.