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

Theodore Andronikos, Alla Sirokofskich

TL;DR

This work studies the PQ penny flip game through a group-theoretic lens, revealing that the game’s core dynamics are captured by the dihedral group $D_8$. It proves that there exist exactly two equivalence classes of winning quantum strategies, each driving the coin along one of two fixed state paths: $(ig|0ig angle,ig|+ig angle,ig|0ig angle)$ and $(ig|0ig angle,ig|-ig angle,ig|0ig angle)$, and that these two paths persist across extensions to all dihedral groups $D_{8n}$ and the full unitary group $U(2)$, provided the rules and the quantum player’s repertoire are as specified. The ambient group is shown to be minimal at $D_8$, and enlarging the action space does not introduce new winning paths. A key result is that, in any extended game, the quantum player must take both the first and the last move to guarantee a sure win against Picard. Collectively, the paper demonstrates that the PQG’s essential structure is robust under broad generalizations, offering a precise group-theoretic understanding of quantum advantage in these games and suggesting directions to explore entanglement and non-basis initial states in future work.

Abstract

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 $D_{8}$. We prove that within $D_{8}$ 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 $D_{8 n}$, $n \geq 1$, 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.

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

TL;DR

This work studies the PQ penny flip game through a group-theoretic lens, revealing that the game’s core dynamics are captured by the dihedral group . It proves that there exist exactly two equivalence classes of winning quantum strategies, each driving the coin along one of two fixed state paths: and , and that these two paths persist across extensions to all dihedral groups and the full unitary group , provided the rules and the quantum player’s repertoire are as specified. The ambient group is shown to be minimal at , and enlarging the action space does not introduce new winning paths. A key result is that, in any extended game, the quantum player must take both the first and the last move to guarantee a sure win against Picard. Collectively, the paper demonstrates that the PQG’s essential structure is robust under broad generalizations, offering a precise group-theoretic understanding of quantum advantage in these games and suggesting directions to explore entanglement and non-basis initial states in future work.

Abstract

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 . We prove that within 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 . We also show that the game can be played in the all dihedral groups , , with any significant change. We examine what happens when Q can draw his moves from the entire 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 at his disposal. We prove that for Q to surely win against Picard, he must make both the first and the last move.

Paper Structure

This paper contains 23 sections, 22 theorems, 75 equations, 15 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Contribution
  4. Organization
  5. Background
  6. The $PQ$ penny flip game
  7. Dihedral groups
  8. The connection between $PQG$ and $D_8$
  9. Matrix representations of rotations and reflections
  10. Orbits and stabilizers
  11. Analyzing Q's strategy in terms of groups
  12. Enlarging the operational space of the game
  13. Inside $D_8$
  14. The smaller dihedral groups $D_3, D_4, D_5, D_6$ and $D_7$
  15. The dihedral groups $D_{8 n}$, $n \geq 1$
  16. ...and 8 more sections

Key Result

Theorem 3.1

The ambient group of the $PQG$ is $D_8$.

Figures (15)

  • Figure 1: The rotational symmetries of the regular heptagon.
  • Figure 2: The reflection symmetries of the regular heptagon.
  • Figure 3: The rotational symmetries of the regular octagon.
  • Figure 4: The reflection symmetries of the regular octagon.
  • Figure 5: Q's Winning strategy in the original $PQG$
  • ...and 10 more figures

Theorems & Definitions (58)

  • Definition 2.1: Winning and dominant strategies
  • Definition 2.2: $PQG$ moves and their composition
  • Definition 2.3
  • Definition 2.4: Group
  • Definition 2.5: Dihedral groups
  • Definition 2.6: Generators
  • Definition 3.1: The ambient group
  • Theorem 3.1: The ambient group of the $PQG$
  • Definition 3.2: Group action
  • Definition 3.3: Orbits and stabilizers
  • ...and 48 more