# Quantum Conditional Strategies and Automata for Prisoners' Dilemmata under the EWL Scheme
## 1 Introduction  
Quantum game theory has gained a lot of research interest since the first pioneering works of the late 1990s [1; 2; 3; 4; 5]. Quantum game theory combines the well-known field of game theory with the emerging branch of computation theory that involves quantum principles. The processing of information can be thought of as a physical phenomenon, and hence, information theory is indissociable from applied and fundamental physics. In many cases, a quantum system description offers certain advantages, at least in theory, compared to the classical situation [6; 7]. In global terms, game theory has been developed to explore the strategic interactions between strategic decision-makers (modeled as players) who try to maximize their payoffs (or minimize their losses) [8; 9]. The theory of games involves the scientific study of situations where there is conflict of interest, and its principles are applied to various frameworks, such as military scenarios, economics, social sciences, and even biology [10; 11; 12; 13]. The theory of games is related to decision theory and can be seen as the branch of decision theory that involves more than one rational decision maker (or \"actor\"). Of course, there are also single-agent decision problems, and they are usually associated with Markovian dynamics. Quantum game theory can be seen as an extension (or a variation) of classical game theory [1; 2; 5; 14]. There is an interesting debate regarding what a quantum game really is and its ontological position in the area of game theory. A useful list of references on this controversy can be found in [14] (citations [2; 3; 4; 5; 6; 7; 8; 9]from that chapter). In the theory of quantum games, one identifies specific characteristics of quantum information and combines them with those of classical multiplayer, non-cooperative games, ultimately looking for differentiation in results related to Nash equilibria, player's strategies, etc. [14; 15], and this is the main reason for someone to characterize this class of games as quantum games. Three primary characteristics distinguish quantum from classical game theory: superposed initial states, quantum entanglement of states, and superposition of actions to be used upon the initial states. [PERSON] [1] and [PERSON] et al. [2] are credited for the first influential works on quantum games considered from the perspective of quantum algorithms. Strategies in classical game theory are either pure (deterministic) or mixed (probabilistic). While not every two-person zero-sum finite game has an equilibrium in the set of pure strategies, [PERSON] showed that there is always an equilibrium at which each player follows a mixed strategy. A mixed strategy deviating from the equilibrium strategy cannot increase a player's expected payoff. [PERSON] demonstrated by the penny-flip game that a player who implements a quantum strategy can increase his expected payoff and explained this result by the efficiency of quantum strategies compared to the classical ones [1]. [PERSON] et al. were the first to propose a quantum extension of the prisoner's dilemma game in [2], which is now commonly referred to as the Eisert-Wilkens-Lewenstein (EWL for short) scheme in the literature. Quantum computing advocates the use of quantum-mechanical methods for computation purposes. Under this perspective, quantum automata have also been thoroughly investigated [16; 17; 18; 19; 20] and have been associated with game theory in [21], where it was shown that they can be related to quantum strategies of repeated games. A similar approach is followed in this paper, this time for more complex settings that require more sophisticated examples of quantum automata. It is evident that such computing machines (i.e., quantum automata) have been shown to outperform classical ones for particular tasks [22; 23; 24] (like the one discussed in this paper), which is an important indicator of the advantages in using similar methods over other already existing ones.