Evolutionarily stable strategies in quantum games

Evolutionarily stable strategy (ESS) in classical game theory is a refinement of Nash equilibrium concept. We investigate the consequences when a small group of mutants using quantum strategies try to invade a classical ESS in a population engaged in symmetric bimatrix game of prisoner's dilemma. Secondly we show that in an asymmetric quantum game between two players an ESS pair can be made to appear or disappear by resorting to entangled or unentangled initial states used to play the game even when the strategy pair remains a Nash equilibrium in both forms of the game.     

Non-factorizable joint probabilities and evolutionarily stable strategies in the quantum prisoner's dilemma game

The well-known refinement of the Nash Equilibrium (NE) called an Evolutionarily Stable Strategy (ESS) is investigated in the quantum Prisoner's Dilemma (PD) game that is played using an Einstein-Podolsky-Rosen type setting. Earlier results report that in this scheme the classical NE remains intact as the unique solution of the quantum PD game. In contrast, we show here that interestingly in this scheme a non-classical solution for the ESS emerges for the quantum PD.     

Stability of Mixed Nash Equilibria in Symmetric Quantum Games

In bimatrix games the Bishop-Cannings theorem of the classical evolutionary game theory does not permitpure evolutionarily stable strategies (ESSs) when a mixed ESS exists. We find the necessary form of two-qubit initialquantum states when a switch-over to a quantum version of the game also changes the evolutionary stability of a mixedsymmetric Nash equilibrium.     

Probabilistic analysis of three-player symmetric quantum games played using the Einstein-Podolsky-Rosen-Bohm setting

This Letter extends our probabilistic framework for two-player quantum games to the multiplayer case, while giving a unified perspective for both classical and quantum games. Considering joint probabilities in the Einstein-Podolsky-Rosen-Bohm (EPR-Bohm) setting for three observers, we use this setting in order to play general three-player noncooperative symmetric games. We analyze how the peculiar non-factorizable joint probabilities provided by the EPR-Bohm setting can change the outcome of a game, while requiring that the quantum game attains a classical interpretation for factorizable joint probabilities. In this framework, our analysis of the three-player generalized Prisoner's Dilemma (PD) shows that the players can indeed escape from the classical outcome of the game, because of non-factorizable joint probabilities that the EPR setting can provide. This surprising result for three-player PD contrasts strikingly with our earlier result for two-player PD, played in the same framework, in which even non-factorizable joint probabilities do not result in escaping from the classical consequence of the game.     

Quantum Game with Restricted Matrix Strategies

We study a quantum game played by two players with restricted multiple strategies. It is found that in this restricted quantum game Nash equilibrium does not always exist when the initial state is entangled. At the same time,we find that when Nash equilibrium exists the payoff function is usually different from that in the classical counterpart except in some special cases. This presents an explicit example showing quantum game and classical game may differ.When designing a quantum game with limited strategies, the allowed strategy should be carefully chosen according to the type of initial state.     

Quantum Game with Restricted Matrix Strategies

We study a quantum game played by two players with restricted multiple strategies. It is found that in this restricted quantum game Nash equilibrium does not always exist when the initial state is entangled. At the same time,we find that when Nasli equilibrium exists the payoff function is usually different from that in the classical counterpart except in some special cases. This presents an explicit example showing quantum game and classical game may differ.When designing a quantum game with limited strategies, the allowed strategy should be carefully chosen according to the type of initial state.     

Maximal Parrondo’s Paradox for Classical and Quantum Markov Chains

Parrondo’s paradox refers to the situation where two, multi-round games with a fixed winning criteria, both with probability greater than one-half for one player to win, are combined. Using a possibly biased coin to determine the rule to employ for each round, paradoxically, the previously losing player now wins the combined game with probability greater than one-half. In this paper, we will analyze classical observed, classical hidden, and quantum versions of a game that displays this paradox. The game we have utilized is simpler than games for which this behavior has been previously noted in the classical and quantum cases. We will show that in certain situations the paradox can occur to a greater degree in the quantum version than is possible in the classical versions.     

Generalized quantum measurement

We overcome one of Bell's objections to ‘quantum measurement’ by generalizing the definition to include systems outside the laboratory. According to this definition, a generalized quantum measurement takes place when the value of a classical variable is influenced significantly by an earlier state of a quantum system. A generalized quantum measurement can then take place in equilibrium systems, provided the classical motion is chaotic. This Letter deals with this classical aspect of quantum measurement, assuming that the Heisenberg cut between the quantum dynamics and the classical dynamics is made at a very small scale. For simplicity, a gas with collisions is modelled by an “Arnold gas”.     

Duality, phase structures, and dilemmas in symmetric quantum games

Symmetric quantum games for 2-player, 2-qubit strategies are analyzed in detail by using a scheme in which all pure states in the 2-qubit Hilbert space are utilized for strategies. We consider two different types of symmetric games exemplified by the familiar games, the Battle of the Sexes (BoS) and the Prisoners’ Dilemma (PD). These two types of symmetric games are shown to be related by a duality map, which ensures that they share common phase structures with respect to the equilibria of the strategies. We find eight distinct phase structures possible for the symmetric games, which are determined by the classical payoff matrices from which the quantum games are defined. We also discuss the possibility of resolving the dilemmas in the classical BoS, PD, and the Stag Hunt (SH) game based on the phase structures obtained in the quantum games. It is observed that quantization cannot resolve the dilemma fully for the BoS, while it generically can for the PD and SH if appropriate correlations for the strategies of the players are provided.     

Potential wells for classical acoustic waves

The acceleration theorem of Bloch waves is utilized to construct random potential wells for classical acoustic waves in systems composed of alternating‘cavities’and‘couplers’.One prominent advantage of this method is these‘cavities’and‘couplers’are all monolayer structures.It allows forming more compact classical potential wells,which leads to the miniaturization of acoustic devices.We systematically investigate properties of harmonic,tangent,hyperbolic function,and square classical potential wells in quasi-periodic superlattices.Results show these classical potential wells are analogues of quantum potential wells.Thus some technologies and concepts in quantum potential well fields may be generalized to classical acoustic wave fields.In addition,some abnormal cases regarding forming classical potential wells are also found.     

Interpreting weak value amplification with a toy realist model

Constructing an ontology for quantum theory is challenging, in part due to measurement back-action. The Aharonov-Albert-Vaidman weak measurement formalism provides a method to predict measurement results (weak values) when back-action is negligible. The weak value appears analogous to a classical conditional mean, yet can be complex and unbounded. We study weak values in the context of a recent quantum optical experiment involving two-photon interactions. The results of the experiment are reinterpreted within a realist ‘stochastic optics’ model of light. We show that the conditional means of the intensities in the model correspond to the experimentally observed weak values and study the breakdown of the model outside the experimentally probed regime in the limit where the weak value predicts ‘anomalous’ results.     

A competitive game whose maximal Nash-equilibrium payoff requires quantum resources for its achievement

While it is known that shared quantum entanglement can offer improved solutions to a number of purely cooperative tasks for groups of remote agents, controversy remains regarding the legitimacy of quantum games in a competitive setting. We construct a competitive game between four players based on the minority game where the maximal Nash-equilibrium payoff when played with the appropriate quantum resource is greater than that obtainable by classical means, assuming a local hidden variable model.     

Coherence stability in three-level systems

We study the decoherence rate for estimating the time at which the coherence instability of a quantum pure state is onset. We analyze the coherence stability of pure states of a three-level quantum system under the effect of a bosonic reservoir and driven by two Raman classical fields. By assuming the boson systems to be in thermal states we find for a symmetric V-system a set of three states free from decoherence and, for a symmetric cascade-system, a two-dimensional subspace whose states are stable against the considered decoherence mechanism.     

Arbiter as the Third Man in Classical and Quantum Games

We study the possible influence of a not necessarily sincere arbiter on the course of classical and quantum 2×2 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 considered as different games.     

Constructing quantum games from symmetric non-factorizable joint probabilities

We construct quantum games from a table of non-factorizable joint probabilities, coupled with a symmetry constraint, requiring symmetrical payoffs between the players. We give the general result for a Nash equilibrium and payoff relations for a game based on non-factorizable joint probabilities, which embeds the classical game. We study a quantum version of Prisoners' Dilemma, Stag Hunt, and the Chicken game constructed from a given table of non-factorizable joint probabilities to find new outcomes in these games. We show that this approach provides a general framework for both classical and quantum games without recourse to the formalism of quantum mechanics.     

Relativistic Quantum Bayesian Game Under Decoherence

We study how Unruh effect and quantum noise affect the payoffs of a quantum conflicting interest Bayesian game. Three types of noisy channels, i.e., the amplitude damping channel, the depolarizing channel and the phase damping channel, are employed to model the decoherence processes. We find that Unruh effect weakens the payoffs in the quantum game and the quantum payoffs are lower than the classical payoffs when the acceleration parameter is large enough. However, the variation of the payoffs with the decoherence parameter is not always monotonic. Sometimes more decoherence may lead to higher payoffs.     

Tripartite Dynamic Zero-Sum Quantum Games

The Nash equilibrium plays a crucial role in game theory. Most of results are based on classical resources. Our goal in this paper is to explore multipartite zero-sum game with quantum settings. We find that in two different settings there is no strategy for a tripartite classical game being fair. Interestingly, this is resolved by providing dynamic zero-sum quantum games using single quantum state. Moreover, the gains of some players may be changed dynamically in terms of the committed state. Both quantum games are robust against the preparation noise and measurement errors.     

Quantum Entanglement in Concept Combinations

Research in the application of quantum structures to cognitive science confirms that these structures quite systematically appear in the dynamics of concepts and their combinations and quantum-based models faithfully represent experimental data of situations where classical approaches are problematical. In this paper, we analyze the data we collected in an experiment on a specific conceptual combination, showing that Bell’s inequalities are violated in the experiment. We present a new refined entanglement scheme to model these data within standard quantum theory rules, where ‘entangled measurements and entangled evolutions’ occur, in addition to the expected ‘entangled states’, and present a full quantum representation in complex Hilbert space of the data. This stronger form of entanglement in measurements and evolutions might have relevant applications in the foundations of quantum theory, as well as in the interpretation of nonlocality tests. It could indeed explain some non-negligible ‘anomalies’ identified in EPR-Bell experiments.     

Experimental realization of quantum games on a quantum computer

We generalize the quantum prisoner's dilemma to the case where the players share a nonmaximally entangled states. We show that the game exhibits an intriguing structure as a function of the amount of entanglement with two thresholds which separate a classical region, an intermediate region, and a fully quantum region. Furthermore this quantum game is experimentally realized on our nuclear magnetic resonance quantum computer.     

Quantum wave processing

In recent years, the concept of quantum computing has arisen as a methodology by which very rapid computations can be achieved. In general, the ‘speed’ of these computations is compared to that of (classical) digital computers, which use sequential algorithms. However, in most quantum computing approaches, the qubits themselves are treated as analog objects. One then needs to ask whether this computational speed-up of the computation is a result of the quantum mechanics, or whether it is due to the nature of the analog structures that are being ‘generated’ for quantum computation? In this paper, we will make two points: (1) quantum computation utilizes analog, parallel computation which often offers no speed advantage over classical computers which are implemented using analog, parallel computation; (2) once this is realized, then there is little advantage in projecting the quantum computation onto the pseudo-binary construct of a qubit. Rather, it becomes more effective to seek the equivalent wave processing that is inherent in the analog, parallel processing. We will examine some wave processing systems which may be useful for quantum computation.