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Acta Physica Sinica

Quantum game— “PQ” problem

Xiao-Kun Yang, Wei Li, Yong-Chang Huang

Published Dec 26, 2023• 0 citations
Publisher

Abstract

This paper reviews the coin flipping problem of “PQ” in a classical two-player game, and shows that when one player adopts quantum strategy, he can beat his classical opponent and gain higher returns. By using quantitative means, the whole “PQ” problem is put under a more general and fair condition, and it is generalized and studied again from many aspects and angles. Finally, we obtain some more essential and important conclusions, and explain the conclusions and its practical significance through some practical examples. At the same time, this paper gives the definition of imperfect fair game and the definition of perfect fair game, revises the quantum coin flipping game to make the game fair, and studies a multi-round version of quantum coin flipping. Some basic conclusions of fair quantum game are obtained, and the meaning of perfectly fair game is explained in practice.

DOI
10.7498/aps.73.20230592

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This paper reviews the coin flipping problem of “PQ” in a classical two-player game, and shows that when one player adopts quantum strategy, he can beat his classical opponent and gain higher returns. By using quantitative means, the whole “PQ” problem is put under a more general and fair condition, and it is generalized and studied again from many aspects and angles. Finally, we obtain some more essential and important conclusions, and explain the conclusions and its practical significance through some practical examples. At the same time, this paper gives the definition of imperfect fair game and the definition of perfect fair game, revises the quantum coin flipping game to make the game fair, and studies a multi-round version of quantum coin flipping. Some basic conclusions of fair quantum game are obtained, and the meaning of perfectly fair game is explained in practice.
This paper reviews the coin flipping problem of PQ in a classical two-player game, and shows that when one player adopts quantum strategy, he can beat his classical opponent and gain higher returns. By means of quantitative means, the whole PQ problem is put in a more general and fair condition, and it is generalized and studied again from many aspects and angles. Finally, we get some more essential and important conclusions, and through some examples to explain the conclusions and its practical significances. At the same time, this paper gives the definitions of imperfect fair games and perfect fair games, revises the quantum coin flipping game to make the game fair, and studies a multi-round version of quantum coin flipping. Some basic conclusions of fair quantum game are obtained, and the meaning of perfectly fair game is explained in practice.
The Quantum Penny Flip is a classic thought experiment in quantum mechanics that demonstrates the weird and counterintuitive nature of the quantum world. To solve the “PQ” problem, we proposed a quantum strategy selection and optimization algorithm, and designed a quantum algorithm for this problem. In terms of practical applications and Case study, this paper reviews the classic two player game – “PQ coin flipping” problem, and shows that when one of the participants adopts quantum strategies, he can defeat his classic opponent and obtain higher returns. By using quantitative methods, the entire “PQ” problem is placed under a more general and fair condition, and it is promoted and studied again from multiple aspects and perspectives. At the same time, this article provides the definitions of “incomplete fair game” and “complete fair game”, modifies the quantum coin flipping game [Meyer, Phys. Rev. lett. 82 (1999) 1052] to make the game fair, and studies the multi round versions of quantum coin flipping. We have drawn some basic conclusions about fair quantum games and explained the meaning of completely fair games in practice. We discussed the limitations and scalability of the “PQ” problem and looked forward to the future development direction of quantum game research.
Abstract Two players of coin-flipping game randomly choose one of two strategies, flipping over or not. Meyer quantized states of the game to be described instead by density matrices of the two-dimensional Hilbert space. In order to obtain and unify generation of the two players coin-flipping game with more than two strategies, we renew the need for equivalently describing Meyer’s PQ coin-flipping game by representing player’s strategies as elements of permutation groups. Expanding the dimension of quantum vector to arbitrary N-dimension in the coin-flipping game model, we generalize Meyer’s PQ coin-flipping game to PQ arbitrary finite strategies game. Owing to the quantum Fourier transform, quantum contrivance of Q has a winning strategy in PQ coin-flipping game with arbitrary finite strategies. Even though the player Q has a small probability of winning his opponent in the classical PQ coin-flipping game with arbitrary finite strategies, he can win his opponent who uses classical mixed strategies by his quantum strategies. Furthermore, we realize the winning quantum strategies of quantum player Q of PQ coin-flipping game with arbitrary finite strategies in macro world.
This paper studies sequential quantum games under the assumption that the moves of the players are drawn from groups and not just plain sets. The extra group structure makes possible to easily derive some very general results, which, to the best of our knowledge, are stated in this generality for the first time in the literature. The main conclusion of this paper is that the specific rules of a game are absolutely critical. The slightest variation may have important impact on the outcome of the game. It is the combination of two factors that determine who wins: (i) the sets of admissible moves for each player, and (ii) the order of moves, i.e., whether the same player makes the first and the last move. Quantum strategies do not a priori prevail over classical strategies. By carefully designing the rules of the game the advantage of either player can be established. Alternatively, the fairness of the game can also be guaranteed.
Coin flipping is a cryptographic primitive in which two spatially separated players, who do not trust each other, wish to establish a common random bit. If we limit ourselves to classical communication, this task requires either assumptions on the computational power of the players or it requires them to send messages to each other with sufficient simultaneity to force their complete independence. Without such assumptions, all classical protocols are so that one dishonest player has complete control over the outcome. If we use quantum communication, on the other hand, protocols have been introduced that limit the maximal bias that dishonest players can produce. However, those protocols would be very difficult to implement in practice because they are susceptible to realistic losses on the quantum channel between the players or in their quantum memory and measurement apparatus. In this paper, we introduce a quantum protocol and we prove that it is completely impervious to loss. The protocol is fair in the sense that either player has the same probability of success in cheating attempts at biasing the outcome of the coin flip. We also give explicit and optimal cheating strategies for both players.
A quantum board game is a multi-round protocol between a single quantum player against the quantum board. Molina and Watrous discovered quantum hedging. They gave an example for perfect quantum hedging: a board game with winning probability < 1, such that the player can win with certainty at least 1-out-of-2 quantum board games played in parallel. Here we show that perfect quantum hedging occurs in a cryptographic protocol – quantum coin flipping. For this reason, when cryptographic protocols are composed in parallel, hedging may introduce serious challenges into their analysis. We also show that hedging cannot occur when playing two-outcome board games in sequence. This is done by showing a formula for the value of sequential two-outcome board games, which depends only on the optimal value of a single board game; this formula applies in a more general setting of possible target functions, in which hedging is only a special case.