This work concentrates on simultaneous move non-cooperating quantum games. Part of it is evidently not new, but it is included for the sake self consistence, as it is devoted to introduction of the mathematical and ph...This work concentrates on simultaneous move non-cooperating quantum games. Part of it is evidently not new, but it is included for the sake self consistence, as it is devoted to introduction of the mathematical and physical grounds of the pertinent topics, and the way in which a simple classical game is modified to become a quantum game (a procedure referred to as a quantization of a classical game). The connection between game theory and information science is briefly stressed, and the role of quantum entanglement (that plays a central role in the theory of quantum games), is exposed. Armed with these tools, we investigate some basic concepts like the existence (or absence) of a pure strategy and mixed strategy Nash equilibrium and its relation with the degree of entanglement. The main results of this work are as follows: 1) Construction of a numerical algorithm based on the method of best response functions, designed to search for pure strategy Nash equilibrium in quantum games. The formalism is based on the discretization of a continuous variable into a mesh of points, and can be applied to quantum games that are built upon two-players two-strategies classical games, based on the method of best response functions. 2) Application of this algorithm to study the question of how the existence of pure strategy Nash equilibrium is related to the degree of entanglement (specified by a continuous parameter γ ). It is shown that when the classical game G<sub>C</sub> has a pure strategy Nash equilibrium that is not Pareto efficient, then the quantum game G<sub>Q</sub> with maximal entanglement (γ = π/2) has no pure strategy Nash equilibrium. By studying a non-symmetric prisoner dilemma game, it is found that there is a critical value 0γ<sub>c</sub> such that for γγ<sub>c</sub> there is a pure strategy Nash equilibrium and for γ≥γ<sub>c </sub>there is no pure strategy Nash equilibrium. The behavior of the two payoffs as function of γ starts at that of the classical ones at (D, D) and approaches the cooperative classical ones at (C, C) (C = confess, D = don’t confess). 3) We then study Bayesian quantum games and show that under certain conditions, there is a pure strategy Nash equilibrium in such games even when entanglement is maximal. 4) We define the basic ingredients of a quantum game based on a two-player three strategies classical game. This requires the introduction of trits (instead of bits) and quantum trits (instead of quantum bits). It is proved that in this quantum game, there is no classical commensurability in the sense that the classical strategies are not obtained as a special case of the quantum strategies.展开更多
Communication complexity is an area of classical computer science which studies how much communication is necessary to solve various distributed computational problems.Quantum information processing can be used to red...Communication complexity is an area of classical computer science which studies how much communication is necessary to solve various distributed computational problems.Quantum information processing can be used to reduce the amount of communication required to carry out some distributed problems.We speak of pseudo-telepathy when it is able to completely eliminate the need for communication.Since it is generally very hard to perfectly implement a quantum winning strategy for a pseudo-telepathy game,quantum players are almost certain to make errors even though they use a winning strategy.After introducing a model for pseudotelepathy games,we investigate the impact of erroneously performed unitary transformations on the quantum winning strategy for the Mermin-GHZ game.The question of how strong the unitary noise can be so that quantum players would still be better than classical ones is also dealt with.展开更多
Quantum pseudo-telepathy(QPT)is a new type of game where the quantum team can win with certainty while the classical one cannot.It means the advantages of quantum participants over classical ones in game.However,there...Quantum pseudo-telepathy(QPT)is a new type of game where the quantum team can win with certainty while the classical one cannot.It means the advantages of quantum participants over classical ones in game.However,there has been no systematic and formal analysis on the QPT game before.Here we present the formal description of the QPT game and the definition of the most simplified QPT.Based on the above definitions,we simplify a famous QPT game,i.e.the Cabllo game.Then,according to some instances,we analyze the minimum best success probability by classical strategies of the two-player QPT,which reflects the advantage of the quantum strategies.Finally,we prove the best success probability by classical strategies for the most simplified QPT is totally related to the number of all possible question combinations.展开更多
文摘This work concentrates on simultaneous move non-cooperating quantum games. Part of it is evidently not new, but it is included for the sake self consistence, as it is devoted to introduction of the mathematical and physical grounds of the pertinent topics, and the way in which a simple classical game is modified to become a quantum game (a procedure referred to as a quantization of a classical game). The connection between game theory and information science is briefly stressed, and the role of quantum entanglement (that plays a central role in the theory of quantum games), is exposed. Armed with these tools, we investigate some basic concepts like the existence (or absence) of a pure strategy and mixed strategy Nash equilibrium and its relation with the degree of entanglement. The main results of this work are as follows: 1) Construction of a numerical algorithm based on the method of best response functions, designed to search for pure strategy Nash equilibrium in quantum games. The formalism is based on the discretization of a continuous variable into a mesh of points, and can be applied to quantum games that are built upon two-players two-strategies classical games, based on the method of best response functions. 2) Application of this algorithm to study the question of how the existence of pure strategy Nash equilibrium is related to the degree of entanglement (specified by a continuous parameter γ ). It is shown that when the classical game G<sub>C</sub> has a pure strategy Nash equilibrium that is not Pareto efficient, then the quantum game G<sub>Q</sub> with maximal entanglement (γ = π/2) has no pure strategy Nash equilibrium. By studying a non-symmetric prisoner dilemma game, it is found that there is a critical value 0γ<sub>c</sub> such that for γγ<sub>c</sub> there is a pure strategy Nash equilibrium and for γ≥γ<sub>c </sub>there is no pure strategy Nash equilibrium. The behavior of the two payoffs as function of γ starts at that of the classical ones at (D, D) and approaches the cooperative classical ones at (C, C) (C = confess, D = don’t confess). 3) We then study Bayesian quantum games and show that under certain conditions, there is a pure strategy Nash equilibrium in such games even when entanglement is maximal. 4) We define the basic ingredients of a quantum game based on a two-player three strategies classical game. This requires the introduction of trits (instead of bits) and quantum trits (instead of quantum bits). It is proved that in this quantum game, there is no classical commensurability in the sense that the classical strategies are not obtained as a special case of the quantum strategies.
基金supported by the research projects MSM0021622419 and 201/0710603
文摘Communication complexity is an area of classical computer science which studies how much communication is necessary to solve various distributed computational problems.Quantum information processing can be used to reduce the amount of communication required to carry out some distributed problems.We speak of pseudo-telepathy when it is able to completely eliminate the need for communication.Since it is generally very hard to perfectly implement a quantum winning strategy for a pseudo-telepathy game,quantum players are almost certain to make errors even though they use a winning strategy.After introducing a model for pseudotelepathy games,we investigate the impact of erroneously performed unitary transformations on the quantum winning strategy for the Mermin-GHZ game.The question of how strong the unitary noise can be so that quantum players would still be better than classical ones is also dealt with.
基金supported by the National Natural Science Foundation of China(Grant Nos.61300181,61272057,61202434,61170270,61100203 and 61121061)Beijing Natural Science Foundation(Grant No.4122054)Beijing Higher Education Young Elite Teacher Project(Grant Nos.YETP0475 and YETP0477)
文摘Quantum pseudo-telepathy(QPT)is a new type of game where the quantum team can win with certainty while the classical one cannot.It means the advantages of quantum participants over classical ones in game.However,there has been no systematic and formal analysis on the QPT game before.Here we present the formal description of the QPT game and the definition of the most simplified QPT.Based on the above definitions,we simplify a famous QPT game,i.e.the Cabllo game.Then,according to some instances,we analyze the minimum best success probability by classical strategies of the two-player QPT,which reflects the advantage of the quantum strategies.Finally,we prove the best success probability by classical strategies for the most simplified QPT is totally related to the number of all possible question combinations.
