Physics Letters A 316 (2003) 317–323 www.elsevier.com/locate/pla Quantum prisoner dilemma under decoherence L.K. Chen a,∗ , Huiling Ang a , D. Kiang a , L.C. Kwek a,b , C.F. Lo c a National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore 639798, Singapore b Quantum Information Technology Project, Department of Physics, National University of Singapore, Kent Ridge Campus, Singapore 119260, Singapore c Department of Physics, Chinese University of Hong Kong, Shatin, Hong Kong, PR China Received 7 December 2002; accepted 31 July 2003 Communicated by A.R. Bishop Abstract It has recently been established that quantum strategies are superior to classical ones for games such as the prisoner’s dilemma. However, quantum states are subject to decoherence. In this Letter, we investigate the effects of decoherence on a quantum game, namely the prisoner dilemma, through three prototype decoherence channels. We show that in the case of prisoner dilemma, the Nash equilibria are not changed by the effects of decoherence for maximally entangled states.  2003 Elsevier B.V. All rights reserved. PACS: 03.67.-a; 03.65.Bz Keywords: Quantum games; Decoherence; Prisoner’s dilemma 1. Introduction The Babylonian Talmud has often been regarded as the first historical instance of a game theoretic prob- lem. This problem baffled Talmudic scholars for many years. However, in 1985, Aumann and Maschler [1] showed that it is indeed a cooperative game. The first theorem in game theory was formulated by Zermelo in 1913 [2]. Since then game theoretic models have been applied extensively to mechanisms behind economi- cal, physical, engineering and biological systems. In economics, game theory has been applied successfully * Corresponding author. E-mail addresses: lkchen@nie.edu.sg (L.K. Chen), dkiang@nie.edu.sg (D. Kiang), lckwek@nie.edu.sg (L.C. Kwek), cflo@phy.cuhk.edu.hk (C.F. Lo). to the study of strategies for handling decision prob- lems [3–5]. In biology, game theoretic models provide a basis for delving into ecological paradigms and the study of wildlife management [6]. Working from an entirely different perspective, physicists have recently shown that it is possible and possibly feasible to apply the laws of quantum me- chanics to information theory. This possibility has raised speculations of an eventual construction of a quantum computer capable of performing fast com- putations through quantum entanglement [7,8]. More- over, it has been shown that it is possible to create unconditionally secure quantum key distribution us- ing quantum information through quantum channels [9,10]. In recent years, there has been a steady interest in the applications of quantum theory to game the- 0375-9601/$ – see front matter  2003 Elsevier B.V. All rights reserved. doi:10.1016/S0375-9601(03)01175-7