LateX, 10 pages, 2 figures, submitted to Physica A special issue (Gene Stanley Conference, Sicily, 2001), v2 minor correction to equations, v3 corrections to results section and table, acknowledgement added
Parrondo's Paradox arises when two losing games are combined to produce a winning one. A history dependent quantum Parrondo game is studied where the rotation operators that represent the toss of a classical biased coin are replaced by general SU(2) operators to transform the game into the quantum domain. In the initial state, a superposition of qubits can be used to couple the games and produce interference leading to quite different payoffs to those in the classical case.
We pursue the possible connections between classical games and quantum computation. The Parrondo game is one in which a random combination of two losing games produces a winning game. We introduce novel realizations of this Parrondo effect in which the player can `win' via random reflections and rotations of the state-vector, and connect these to known quantum algorithms.
Earlier work on the quantization of the history dependent (HD) Parrondo game by Flitney, Ng, and Abbott led to the FNA protocol. We propose an alternative quantization protocol for this game which differs from the FNA protocol in various aspects.
Coordination and cooperation are among the most important issues of game theory. Recently, the attention turned to game theory on graphs and social networks. Encouraged by interesting results obtained in quantum evolutionary game analysis, we study cooperative Parrondo's games in a quantum setup. The game is modeled using multidimensional quantum random walks with biased coins. We use the GHZ and W entangled states as the initial state of the coins. Our analysis shows than an apparent paradox in cooperative quantum games and some interesting phenomena can be observed.
We present a quantum implementation of Parrondo's game with randomly switched strategies using 1) a quantum walk as a source of ``randomness'' and 2) a completely positive (CP) map as a randomized evolution. The game exhibits the same paradox as in the classical setting where a combination of two losing strategies might result in a winning strategy. We show that the CP-map scheme leads to significantly lower net gain than the quantum walk scheme.
We study a quantum walk in one-dimension using two different "coin" operators. By mixing two operators, both of which give a biased walk with negative expectation value for the walker position, it is possible to reverse the bias through interference effects. This effect is analogous to that in Parrondo's games, where alternating two losing (gambling) games can produce a winning game. The walker bias is produced by introducing a phase factor into the coin operator, with two different phase factors giving games $A$ and $B$. We give the range of phases for which the Parrondo effect can be obtaine…
A quantum logic gate of particular interest to both electrical engineers and game theorists is the quantum multiplexer. This shared interest is due to the facts that an arbitrary quantum logic gate may be expressed, up to arbitrary accuracy, via a circuit consisting entirely of variations of the quantum multiplexer, and that certain one player games, the history dependent Parrondo games, can be quantized as games via a particular variation of the quantum multiplexer. However, to date all such quantizations have lacked a certain fundamental game theoretic property. The main result in this disse…
Parrondo's paradox is a well-known counterintuitive phenomenon, where the combination of unfavorable situations can establish favorable ones. In this paper, we study one-dimensional discrete-time quantum walks, manipulating two different coins (two-state) operators representing two losing games A and B, respectively, to create the Parrondo effect in the quantum domain. We exhibit that games A and B are losing games when played individually but could produce a winning expectation when played alternatively for a particular sequence of different periods for distinct choices of the relative phase.…
We present a new form of a Parrondo game using discrete-time quantum walk on a line. The two players A and B with different quantum coins operators, individually losing the game can develop a strategy to emerge as joint winners by using their coins alternatively, or in combination for each step of the quantum walk evolution. We also present a strategy for a player A (B) to have a winning probability more than player B (A). Significance of the game strategy in information theory and physical applications are also discussed.
We discuss the effect of correlated noise on the robustness of quantum coherent phenomena. First we consider a simple, toy model to illustrate the effect of such correlations on the decoherence process. Then we show how decoherence rates can be suppressed using a Parrondo-like effect. Finally, we report the results of many-body calculations in which an experimentally-measurable quantum coherence phenomenon is significantly enhanced by non-Markovian dynamics arising from the noise source.
In the context of quantum information theory, "quantization" of various mathematical and computational constructions is said to occur upon the replacement, at various points in the construction, of the classical randomization notion of probability distribution with higher order randomization notions from quantum mechanics such as quantum superposition with measurement. For this to be done "properly", a faithful copy of the original construction is required to exist within the new "quantum" one, just as is required when a function is extended to a larger domain. Here procedures for extending hi…
We consider the discrete-time quantum walk in the plane, and present a quantum implementation of Parrondo's game for four players. Physical significance of the game strategies are also discussed.
Parrondo games are coin flipping games with the surprising property that alternating plays of two losing games can produce a winning game. We show that this phenomenon can be modelled by probabilistic lattice gas automata. Furthermore, motivated by the recent introduction of quantum coin flipping games, we show that quantum lattice gas automata provide an interesting definition for quantum Parrondo games.
We construct a Parrondo's game using discrete time quantum walks. Two lossing games are represented by two different coin operators. By mixing the two coin operators $U_{A}(α_{A},β_{A},γ_{A})$ and $U_{B}(α_{B},β_{B},γ_{B})$, we may win the game. Here we mix the two games in position instead of time. With a number of selections of the parameters, we can win the game with sequences ABB, ABBB, \emph{et al}. If we set $β_{A}=45^{\circ},γ_{A}=0,α_{B}=0,β_{B}=88^{\circ}$, we find the game 1\emph{}with {\normalsize $U_{A}^{S}=U^{S}(-51^{\circ},45^{\circ},0)$, $U_{B}^{S}=U^{S}(0,88^{\circ},-16^{\circ}…
We study the effect of quantum noise on history dependent quantum Parrondo's games by taking into account different noise channels. Our calculations show that entanglement can play a crucial role in quantum Parrondo's games. It is seen that for the maximally entangled initial state in the presence of decoherence, the quantum phases strongly influence the payoffs for various sequences of the game. The effect of amplitude damping channel leads to winning payoffs. Whereas the depolarizing and phase damping channels lead to the losing payoffs. In case of amplitude damping channel, the payoffs are …
Playing a Parrondo's game with a qutrit is the subject of this paper. We show that a true quantum Parrondo's game can be played with a 3 state coin(qutrit) in a 1D quantum walk in contrast to the fact that playing a true Parrondo's game with a 2 state coin(qubit) in 1D quantum walk fails in the asymptotic limits.
We propose a quantum implementation of a capital-dependent Parrondo's paradox that uses $O(\log_2(n))$ qubits, where $n$ is the number of Parrondo games. We present its implementation in the quantum computer language (QCL) and show simulation results.
A quantum walker moves on the integers with four extra degrees of freedom, performing a coin-shift operation to alter its internal state and position at discrete units of time. The time evolution is described by a unitary process. We focus on finding the limit probability law for the position of the walker and study it by means of Fourier analysis. The quantum walker exhibits both localization and a ballistic behavior. Our two results are given as limit theorems for a 2-period time-dependent walk and they describe the location of the walker after it has repeated the unitary process a large num…
Parrondo's paradox is ubiquitous in games, ratchets and random walks.The apparent paradox, devised by J.~M.~R.~Parrondo, that two losing games $A$ and $B$ can produce an winning outcome has been adapted in many physical and biological systems to explain their working. However, proposals on demonstrating Parrondo's paradox using quantum walks failed {for large number of steps}. In this work, we show that instead of a single coin if we consider a two coin initial state which may or may not be entangled, we can observe a genuine Parrondo's paradox with quantum walks. Further we focus on reasons f…
There exist memoryless zero-capacity quantum channels that when used jointly result in the channel with positive capacity. This phenomenon is called superactivation. Making use of Parrondo's paradox, we exhibit examples of superactivation-like effect for the capacity of classical communication channels as well as quantum and private capacity of quantum channels with memory. There are several ingredients necessary for superactivation of quantum capacity to occur in memoryless case. The first one is the requirement for the quantum channels which are amenable for superactivation to come from two …
We show that a genuine Parrondo paradox can emerge in two-state quantum walks without resorting to experimentally intricate high-dimensional coins. To achieve such goal we employ a time-dependent coin operator without breaking the translation spatial invariance of the system.
We investigate the possibility of implementing a sequence of quantum walks whose probability distributions give an overall positive winning probability, while it is negative for the single walks (Parrondo's paradox). In particular, we have in mind an experimental realisation with a Bose-Einstein condensate in which the walker's space is momentum space. Experimental problems in the precise implementation of the coin operations for our discrete-time quantum walks are analysed in detail. We study time-dependent phase fluctuations of the coins as well as perturbations arising from the finite momen…
Geometric phase plays an important role in evolution of pure or mixed quantum states. However, when a system undergoes decoherence the development of geometric phase may be inhibited. Here, we show that when a quantum system interacts with two competing environments there can be enhancement of geometric phase. This effect is akin to Parrondo like effect on the geometric phase which results from quantum frustration of decoherence. Our result suggests that the mechanism of two competing decoherence can be useful in fault-tolerant holonomic quantum computation.
Quantum entanglement has multiple applications in quantum information processing. Developing methods to generate highly entangled states independent of initial conditions is an essential task. Herein we aim to generate highly entangled states via discrete-time quantum walks. We propose deterministic Parrondo sequences that generate states that are generally much more entangled than states produced by sequences using only one of the two coins. We show that some Parrondo sequences generate highly entangled states, which are independent of the phase of the initial state used and further lead to m…
The Geometry of Transport in Quantum Walks and Parrondo's Paradox
Instituto de Física, Universidad Nacional Autónoma de México（墨西哥国立自治大学物理研究所）；Escuela de Ciencias Físicas y Matemáticas, Universidad de San Carlos de Guatemala（危地马拉圣卡洛斯大学物理与数学院）；Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien（维也纳量子科学与技术中心，原子物理研究所，维也纳工业大学）
We introduce a method of analyzing entanglement enhanced quantum games on regular lattices of agents. Our method is valid for setups with periodic and non-periodic boundary conditions. To demonstrate our approach we study two different types games, namely the prisoner's dilemma game and a cooperative Parrondo's game. In both cases we obtain results showing, that entanglement is a crucial resource necessary for the agents to achieve positive capital gain.
A framework for discussing relationships between different types of games is proposed. Within the framework, quantum simultaneous games, finite quantum simultaneous games, quantum sequential games, and finite quantum sequential games are defined. In addition, a notion of equivalence between two games is defined. Finally, the following three theorems are shown: (1) For any quantum simultaneous game G, there exists a quantum sequential game equivalent to G. (2) For any finite quantum simultaneous game G, there exists a finite quantum sequential game equivalent to G. (3) For any finite quantum se…
The noncooperative Nash equilibrium solution of classical games corresponds to a rational expectations attitude on the part of the players. However, in many cases, games played by human players have outcomes very different from Nash equilibria. A restricted version of quantum games is proposed to implement, in mathematical games, the interplay of self-interest and internalized social norms that rules human behavior.
An example of the macroscopic game of two partners consisting of two classical games played simultaneously with special dependence of strategies is considered. The average profit of each partner is equal to the average profit obtained in the quantum game with two noncommuting operators for the spin one half system with strategies defined by the wave function. Nash equilibria in the macroscopic game coincide with Nash equilibria of the quantum game.
We study possible influence of not necessarily sincere arbiter on the course of classical and quantum 2x2 games and we show that this influence in the quantum case is much bigger than in the classical case. Extreme sensitivity of quantum games on initial states of quantum objects used as carriers of information in a game shows that a quantum game, contrary to a classical game, is not defined by a payoff matrix alone but also by an initial state of objects used to play a game. Therefore, two quantum games that have the same payoff matrices but begin with different initial states should be consi…
Several variants of nonlocal games have been considered in the study of quantum entanglement and nonlocality. This paper concerns two of these variants, called quantum-classical games and extended nonlocal games. We give a construction of an extended nonlocal game from any quantum-classical game that allows one to translate certain facts concerning quantum- classical games to extended nonlocal games. In particular, based on work of Regev and Vidick, we conclude that there exist extended nonlocal games for which no finite-dimensional entangled strategy can be optimal. While this conclusion is a…
A game-theoretic setting provides a mathematical basis for analysis of strategic interaction among competing agents and provides insights into both classical and quantum decision theory and questions of strategic choice. An outstanding mathematical question, is to understand the conditions under which a classical game-theoretic setting can be transformed to a quantum game, and under which conditions there is an equivalence. In this paper, we consider quantum games as those that allow non-factorizable probabilities. We discuss two approaches for obtaining a non-factorizable game and study the o…
Quantum pseudo-telepathy games are good examples of explaining the strangeness of quantum mechanics and demonstrating the advantage of quantum resources over classical resources. Most of the quantum pseudo-telepathy games are common interest games, nevertheless conflicting interest games are more widely used to model real world situations. Recently Pappa et al. (Phys. Rev. Lett. 114, 020401, 2015) proposed the first two-party conflicting interest game where quantum advice enhances social optimality. In the present paper we give two new three-party conflicting interest games and show that quant…
We pursue a general theory of quantum games. We show that quantum games are more efficient than classical games, and provide a saturated upper bound for this efficiency. We demonstrate that the set of finite classical games is a strict subset of the set of finite quantum games. We also deduce the quantum version of the Minimax Theorem and the Nash Equilibrium Theorem.
We present a perspective on quantum games that focuses on the physical aspects of the quantities that are used to implement a game. If a game is to be played, it has to be played with objects and actions that have some physical existence. We call such games playable. By focusing on the notion of playability for games we can more clearly see the distinction between classical and quantum games and tackle the thorny issue of what it means to quantize a game. The approach we take can more properly be thought of as gaming the quantum rather than quantizing a game and we find that in this perspectiv…
Quantum games have proposed a new point of view for the solution of the classical problems and dilemmas in game theory. Certain quantization relationships can be proposed with the objective that a game can be generalized into a quantum domain where the linear superposition of actions is allowed. This quantization let us describe and solution problems originated by conflicting or cooperative behaviors among the members of a system from the point of view of quantum mechanical interactions. This leads us to interesting aspects which only can be observed through the quantization of a game like the…
We quantize prisoners dilemma and chicken game by our generalized quantization scheme to explore the role of quantum discord in quantum games. In order to establish this connection we use Werner-like state as an initial state of the game. In this quantization scheme measurement can be performed in entangled as well as in product basis. For the measurement in entangled basis the dilemma in both the games can be resolved by separable states with non-zero quantum discord. Similarly for product basis measurement the payoffs are quantum mechanical only for nonzero values of quantum discord.
We use the example of playing a 2-player game with entangled quantum objects to investigate the effect of quantum correlation. We find that for simple game scenarios it is classical correlation that is the central feature and that these simple quantum games are not sensitive to the quantum part of the correlation. In these games played with quantum objects it is possible to transform a game such as Prisoner's Dilemma into the game of Chicken. We show that this behaviour, and the associated enhanced equilibrium payoff over playing the game with quantum objects in non-entangled states, is entire…
A quantum version of the Matching Pennies (MP) game is proposed that is played using an Einstein-Podolsky-Rosen-Bohm (EPR-Bohm) setting. We construct the quantum game without using the state vectors, while considering only the quantum mechanical joint probabilities relevant to the EPR-Bohm setting. We embed the classical game within the quantum game such that the classical MP game results when the quantum mechanical joint probabilities become factorizable. We report new Nash equilibria in the quantum MP game that emerge when the quantum mechanical joint probabilities maximally violate the Clau…
Categorical quantum mechanics, which examines quantum theory via dagger-compact closed categories, gives satisfying high-level explanations to the quantum information procedures such as Bell-type entanglement or complementary observables (\cite{AC}, \cite{Co}, \cite{Co2}). Inspired by the fact that Quantum Game Theory can be seen as branch of quantum information, we express Quantum Game Theory procedures using the topological semantics provided by Categorical Quantum Mechanics. We also investigate Bayesian Games with correlation from this novel point of view while considering the connection be…
The physical world obeys the rules of quantum, as opposed to classical, physics. Since the playing of any particular game requires physical resources, the question arises as to how Game Theory itself would change if it were extended into the quantum domain. Here we provide a general formalism for {\em quantum} games, and illustrate the explicit application of this new formalism to a quantized version of the well-known prisoner's dilemma game.
In this work we propose and develop modified quantum games (zero and non-zero sum) in which payoffs and strategies are entangled. For the games studied, Nash and Pareto equilibriums are always obtained indicating that there are some interesting cases where quantum games can be applied.
Quantum guessing games form a versatile framework for studying different tasks of information processing. A quantum guessing game with posterior information uses quantum systems to encode messages and classical communication to give partial information after a quantum measurement has been performed. We present a general framework for quantum guessing games with posterior information and derive structure and reduction theorems that enable to analyze any such game. We formalize symmetry of guessing games and characterize the optimal measurements in cases where the symmetry is related to an irred…
The game of Prisoner Dilemma is analyzed to study the role of measurement basis in quantum games. Four different types of payoffs for quantum games are identified on the basis of different combinations of initial state and measurement basis. A relation among these different payoffs is established.
