Quantum Game of Two Discriminable Coins

Ren, Heng-Feng; Wang, Qingliang

doi:10.1007/s10773-007-9625-6

Cited by 14 publications

(13 citation statements)

References 9 publications

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“…Wang et al [7] also generalized the coin tossing game to an n-state game. Ren et al [8] developed specific methods that enabled them to solve the problem of quantum coin-tossing in a roulette game. Specifically, they used two methods, which they called analogy and isolation methods respectively, to tackle the above problem.…”

Section: Related Workmentioning

confidence: 99%

Finite Automata Capturing Winning Sequences for All Possible Variants of the PQ Penny Flip Game

et al. 2018

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Abstract:The meticulous study of finite automata has produced many important and useful results. Automata are simple yet efficient finite state machines that can be utilized in a plethora of situations. It comes, therefore, as no surprise that they have been used in classic game theory in order to model players and their actions. Game theory has recently been influenced by ideas from the field of quantum computation. As a result, quantum versions of classic games have already been introduced and studied. The PQ penny flip game is a famous quantum game introduced by Meyer in 1999. In this paper, we investigate all possible finite games that can be played between the two players Q and Picard of the original PQ game. For this purpose, we establish a rigorous connection between finite automata and the PQ game along with all its possible variations. Starting from the automaton that corresponds to the original game, we construct more elaborate automata for certain extensions of the game, before finally presenting a semiautomaton that captures the intrinsic behavior of all possible variants of the PQ game. What this means is that, from the semiautomaton in question, by setting appropriate initial and accepting states, one can construct deterministic automata able to capture every possible finite game that can be played between the two players Q and Picard. Moreover, we introduce the new concepts of a winning automaton and complete automaton for either player.

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Section: Related Workmentioning

confidence: 99%

Finite Automata Capturing Winning Sequences for All Possible Variants of the PQ Penny Flip Game

et al. 2018

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“…Hence, their axes of symmetry intersect in an angle π 8 , as shown in Figure 4. Moreover, their product FH, which is given in (3), is just the rotator R 2π 8 , as can be verified by employing Formula (7). Therefore, by invoking the presentation (6), associating s to F, and t to H, or vice versa, it becomes evident that F and H generate the dihedral group D 8 .…”

Section: The Connection Between Pqg and D 8 31 Matrix Representations Of Rotations And Reflectionsmentioning

confidence: 76%

“…Afterwards, many researchers generalized this game to n-dimensional quantum systems. Important results in this direction were obtained by [6][7][8]. These results indicated that under a specific set of rules, the quantum player does have an advantage over the classical player.…”

Section: Related Workmentioning

confidence: 95%

The Connection between the PQ Penny Flip Game and the Dihedral Groups

Andronikos

Sirokofskich

2021

Mathematics

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This paper is inspired by the PQ penny flip game. It employs group-theoretic concepts to study the original game and its possible extensions. In this paper, it is shown that the PQ penny flip game can be associated, in a precise way, with the dihedral group D8 and that within D8 there exist precisely two classes of equivalent winning strategies for Q. This is achieved by proving that there are exactly two different sequences of states that can guarantee Q’s win with probability 1.0. It is demonstrated that the game can be played in every dihedral group D8n, where n≥1, without any significant change. A formal examination of what happens when Q can draw their moves from the entire U(2), leads to the conclusion that, again, there are exactly two classes of winning strategies for Q, each class containing an infinite number of equivalent strategies, but all of them sending the coin through the same sequence of states as before. Finally, when general extensions of the game, with the quantum player having U(2) at their disposal, are considered, a necessary and sufficient condition for Q to surely win against Picard is established: Q must make both the first and the last move in the game.

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“…Afterwards, many researchers generalized this game to n-dimensional quantum systems. Important results in this direction were obtained by [6], [7], and [8]. These results indicated that under a specific set of rules, the quantum player does have an advantage over the classical player.…”

Section: Related Workmentioning

confidence: 91%

The connection between the $PQ$ penny flip game and the dihedral groups

Andronikos¹,

Sirokofskich²

2021

Preprint

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This paper is inspired by the PQ penny flip game. It employs group-theoretic concepts to study the original game and also its possible extensions. We show that the PQ penny flip game can be associated with the dihedral group D8. We prove that within D8 there exist precisely two classes of winning strategies for Q. We establish that there are precisely two different sequences of states that can guaranteed Q's win with probability 1.0. We also show that the game can be played in the all dihedral groups D8n, n ≥ 1, with any significant change. We examine what happens when Q can draw his moves from the entire U (2) and we conclude that again, there are exactly two classes of winning strategies for Q, each class containing now an infinite number of equivalent strategies, but all of them send the coin through the same sequence of states as before. Finally, we consider general extensions of the game with the quantum player having U (2) at his disposal. We prove that for Q to surely win against Picard, he must make both the first and the last move.

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Quantum Game of Two Discriminable Coins

Cited by 14 publications

References 9 publications

Finite Automata Capturing Winning Sequences for All Possible Variants of the PQ Penny Flip Game

Finite Automata Capturing Winning Sequences for All Possible Variants of the PQ Penny Flip Game

The Connection between the PQ Penny Flip Game and the Dihedral Groups

The connection between the $PQ$ penny flip game and the dihedral groups

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