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Toward a general theory of quantum games

Gus Gutoski, John Watrous

Published Jan 1, 2006• 143 citations
OpenAlex

Abstract

Abstract We study properties of quantum strategies, which are complete specifications of a givenparty's actions in any multiple-round interaction involving the exchange of quantum information with one or more other parties. In particular, we focus on a representation of quantumstrategies that generalizes the Choi-Jamiol/kowski representation of quantum operations. This new representation associates with each strategy a positive semidefinite operator acting onlyon the tensor product of its input and output spaces. Various facts about such representations are established, and two applications are discussed: the first is a new and conceptually simpleproof of Kitaev's lower bound for strong coin-flipping, and the second is a proof of the exact characterization QRG = EXP of the class of problems having quantum refereed games. 1 Introduction The theory of games provides a general structure within which both cooperation and competitionamong independent entities may be modeled, and provides powerful tools for analyzing these models. Applications of this theory have fundamental importance in many areas of science.This paper considers games in which the players may exchange and process quantum information. We focus on competitive games, and within this context the types of games we consider arevery general. For instance, they allow multiple rounds of interaction among the players involved, and place no restrictions on players ' strategies beyond those imposed by the theory of quantuminformation. While classical games can be viewed as a special case of quantum games, it is important tostress that there are fundamental differences between general quantum games and classical games. For example, the two most standard representations of classical games, namely the normal formand extensive form representations, are not directly applicable to general quantum games. This is due to the nature of quantum information, which admits a continuum of pure (meaning extremal)

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Abstract We study properties of quantum strategies, which are complete specifications of a givenparty's actions in any multiple-round interaction involving the exchange of quantum information with one or more other parties. In particular, we focus on a representation of quantumstrategies that generalizes the Choi-Jamiol/kowski representation of quantum operations. This new representation associates with each strategy a positive semidefinite operator acting onlyon the tensor product of its input and output spaces. Various facts about such representations are established, and two applications are discussed: the first is a new and conceptually simpleproof of Kitaev's lower bound for strong coin-flipping, and the second is a proof of the exact characterization QRG = EXP of the class of problems having quantum refereed games. 1 Introduction The theory of games provides a general structure within which both cooperation and competitionamong independent entities may be modeled, and provides powerful tools for analyzing these models. Applications of this theory have fundamental importance in many areas of science.This paper considers games in which the players may exchange and process quantum information. We focus on competitive games, and within this context the types of games we consider arevery general. For instance, they allow multiple rounds of interaction among the players involved, and place no restrictions on players ' strategies beyond those imposed by the theory of quantuminformation. While classical games can be viewed as a special case of quantum games, it is important tostress that there are fundamental differences between general quantum games and classical games. For example, the two most standard representations of classical games, namely the normal formand extensive form representations, are not directly applicable to general quantum games. This is due to the nature of quantum information, which admits a continuum of pure (meaning extremal)
We study properties of quantum strategies, which are complete specifications of a given party's actions in any multiple-round interaction involving the exchange of quantum information with one or more other parties. In particular, we focus on a representation of quantum strategies that generalizes the Choi-Jamiołkowski representation of quantum operations. This new representation associates with each strategy a positive semidefinite operator acting only on the tensor product of its input and output spaces. Various facts about such representations are established, and two applications are discussed: the first is a new and conceptually simple proof of Kitaev's lower bound for strong coin-flipping, and the second is a proof of the exact characterization QRG = EXP of the class of problems having quantum refereed games.
This thesis is divided into two parts. In Part I we introduce a new formalism for quantum strategies, which specify the actions of one party in any multi-party interaction involving the exchange of multiple quantum messages among the parties. This formalism associates with each strategy a single positive semidefinite operator acting only upon the tensor product of the input and output message spaces for the strategy. We establish three fundamental properties of this new representation for quantum strategies and we list several applications, including a quantum version of von Neumann's celebrated 1928 Min-Max Theorem for zero-sum games and an efficient algorithm for computing the value of such a game. In Part II we establish several properties of a class of quantum operations that can be implemented locally with shared quantum entanglement or classical randomness. In particular, we establish the existence of a ball of local operations with shared randomness lying within the space spanned by the no-signaling operations and centred at the completely noisy channel. The existence of this ball is employed to prove that the weak membership problem for local operations with shared entanglement is strongly NP-hard. We also provide characterizations of local operations in terms of linear functionals that are positive and "completely" positive on a certain cone of Hermitian operators, under a natural notion of complete positivity appropriate to that cone. We end the thesis with a discussion of the properties of no-signaling quantum operations.
We develop a method for the transfer of perfect strategies between various classes of two-player, one round cooperative non-local games with quantum inputs and outputs using the simulation paradigm in quantum information theory. We show that such a transfer is possible when canonically associated operator spaces for each game are quantum homomorphic or isomorphic, as defined in the joint work of H. and Todorov (2024). We examine a new class of QNS correlations, needed for the transfer of strategies between games, and characterize them in terms of states on tensor products of canonical operator systems. We define jointly tracial correlations and show they correspond to traces acting on tensor products of canonical C∗{\rm C}^{*}C∗-algebras associated with individual game parties. We then make an inquiry into the initial application of such results to the study of concurrent quantum games.
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.
Game theory is the study of strategic competition used in many branches of science. In general, a game is a match between two or more players and each wants to maximize his or her score. In a game, the player’s score depends on their strategies. The goal is to find the optimized decision, with regard to the payoff, in conformity with the rules of the game. This thesis considers the games with two players who know both their own and their rival’s payoff. However, they do not know the strategy chosen by the opponent.\nIn a classical game, the options of players are referred to as cooperation and defect. Quantum game theory is a generalization of classical game theory. In quantum games a linear combination of two options specify the strategy based on qubit. The quantum domain is characterized by entanglement. The quantum entanglement coefficient (y) varies from 0 to . The game is in the classical regime when y is smaller than a threshold. Quantum entanglement is intrinsically non-local.\nIn an iterated game tournament, 63 different strategies contest with four known strategies. Depending on the result, they are divided to a number of groups. The competition specifies which group of strategies can obtain a higher score.\nOther types of games are discussed in the thesis. They are known as incomplete information games. In these games one player has two payoff matrixes, and the rival does not know which of them is used. We discuss the advantage of using quantum strategy over the classical game, particular in avoiding the dilemma in classical games.
This paper is concerned with complexity theoretic aspects of a general formulation of quantum game theory that models strategic interactions among rational agents that process and exchange quantum information. In particular, we prove that the computational problem of finding an approximate Nash equilibrium in a broad class of quantum games is, like the analogous problem for classical games, included in (and therefore complete for) the complexity class PPAD. Our main technical contribution, which facilitates this inclusion, is an extension of prior methods in computational game theory to strategy spaces that are characterized by semidefinite programs.