The Principle of Minimal and Maximal Action in the QRG Model

V. V. Velichenko

Wuniu Liu, Zhihui Li, Yongming Li

The reachability of quantum games refers to the ability of two players to reach a certain subspace of Hilbert spaces in a quantum game modeled by a newly proposed quantum game process (QGP). We refer to this as quantum reachability games (QRGs). Reachability analysis is crucial in studying the strategic behavior and decision-making processes of players in quantum games, as it aids in devising optimal strategies to achieve desired objectives. In a QRG, players make decisions based on their quantum strategies related to decision epochs, which are represented by quantum operations or quantum measurements. We present an algorithm for evaluating two-player strategies, and investigate the optimal QRG for two players, where the goal of player 1 is to maximize his/her probability of reaching a given target subspace, while player 2 aims to minimize it. We show that any optimal QRG are determined, i.e., both of the two players have the same probability of winning in non-cooperative reachability games. Furthermore, our results also demonstrate that reachability games in the quantum world can be undecidable in certain cases. These findings are significantly distinct from those in the probabilistic or classical scenarios. Finally, we present three case studies, penny flipover, ambushing quantum walk and peer-to-peer scheme game on quantum network, to illustrate the potential applications and implications of QRGs.

Walter Dittrich

Graeme Heald

M. C. Smith-Agreda

Fernando Shinoske Tagawa de Lemos Pires, Afonso Werner da Rosa, Nathan Willig Lima +1 more

In the present work we present a didactic reconstruction of de Broglie’s demonstration of the equivalence between two fundamental physical concepts: action minimization and entropy maximization. The demonstration was first developed in de Broglie’s Thermodynamics of the Isolated Particle (an attempt to develop a causal interpretation of Quantum Theory in the 1960s). In this work, de Broglie argues from a particle in a hypothetical sub-quantum medium, which operates as a kind of hidden thermostat exchanging energy with the particle. In the discussion about the action minimization and entropy maximization, however, there is not any reference to the quantum nature of the system. So, it is a demonstration valid for any system in a thermal bath in some usual thermostat. Louis de Broglie concludes that the “natural” trajectory of a particle is not only the one that minimizes the action of the particle but also maximizes the entropy of the thermostat. Our objective is to reconstruct and restructure de Broglie’s original demonstration in order to rescue an historical debate, that has not been addressed in the literature, and that is of interest to contemporary fields such as non-equilibrium thermodynamics.