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Advances in transdisciplinary engineering

Fair Game of Quantum Penny Flip

Xiaoken Yang, Wei Li, Yong-Chang Huang

Published Oct 17, 2023• 0 citations
Publisher

Abstract

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.

DOI
10.3233/atde230561

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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.
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.
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.
Games with unawareness model strategic situations in which players’ perceptions about the game are limited. They take into account the fact that the players may be unaware of some of the strategies available to them or their opponents as well as the players may have a restricted view about the number of players participating in the game. The aim of the research is to introduce this notion into the theory of quantum games. We shall focus on PQ Penny Flip game introduced by D. Meyer. We shall formalize the previous results and consider other cases of unawareness in the game.
In this paper, we formulate and analyze generalizations of the quantum penny flip game. In the penny flip game, one coin has two states, heads or tails, and two players apply alternating operations on the coin. In the original Meyer game, the first player is allowed to use quantum (i.e., non-commutative) operations, but the second player is still only allowed to use classical (i.e., commutative) operations. In our generalized games, both players are allowed to use non-commutative operations, with the second player being partially restricted in what operators they use. We show that even if the second player is allowed to use “phase-variable” operations, which are non-Abelian in general, the first player still has winning strategies. Furthermore, we show that even when the second player is allowed to choose one from two or more elements of the group |U(2)U(2)U(2)|⁠, the second player has winning strategies under certain conditions. These results suggest that there is often a method for restoring the quantum state disturbed by another agent.
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.
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.