A necessary and sufficient condition to play games in quantum mechanical settings

S K Özdemir,J Shimamura,N Imoto

doi:10.1088/1367-2630/9/2/043

Abstract

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.

Highlights

Mathematical models and techniques of game theory have increasingly been used by computer and information scientists, i.e., distributed computing, cryptography, watermarking and information hiding tasks can be modelled as games [1, 2, 3, 4, 5, 6]
Since physical systems, which are governed by the laws of quantum mechanics, are used during information flow, game theory becomes closely related to quantum mechanics, physics, computation and information sciences
This effort, has been criticized on the basis of using artificial models [18, 19], has produced significant results: (i) Dilemmas in some games can be resolved [20, 9, 16, 15, 21, 22], (ii) playing quantum games can be more efficient in terms of communication cost; less information needs to be exchanged in order to play the quantized versions of classical games [17, 8, 16], (iii) entanglement is not necessary for the emergence of Nash Equilibrium but for obtaining the highest possible sum of payoffs [16], and (iv) quantum advantage does not survive in the presence of noise above a critical level [23, 24]

Summary

Introduction

Mathematical models and techniques of game theory have increasingly been used by computer and information scientists, i.e., distributed computing, cryptography, watermarking and information hiding tasks can be modelled as games [1, 2, 3, 4, 5, 6]. Quantum mechanics is introduced into game theory through the use of quantum bits (qubits) instead of classical bits, quantum operations and entanglement which is a quantum correlation with a highly complex structure and is considered to be the essential ingredient to exploit the potential power of quantum information processing This effort, has been criticized on the basis of using artificial models [18, 19], has produced significant results: (i) Dilemmas in some games can be resolved [20, 9, 16, 15, 21, 22], (ii) playing quantum games can be more efficient in terms of communication cost; less information needs to be exchanged in order to play the quantized versions of classical games [17, 8, 16], (iii) entanglement is not necessary for the emergence of Nash Equilibrium but for obtaining the highest possible sum of payoffs [16], and (iv) quantum advantage does not survive in the presence of noise above a critical level [23, 24]. An important consequence of this criterion in game theory is the main contribution of this study: Derivation of the necessary and sufficient condition for entangled states and quantum operators that can be used in the quantized versions of classical games

Definitions and model

Classification of N-player two-strategy games

Reproducibility criterion to play games in quantum mechanical settings

Reproducibility criterion as a benchmark

Conclusion