arXiv subjects

Marek Szopa

Publications and source records attributed to Marek Szopa.

7 recordsLinked to original sources

arXiv2025

QEGS: A Mathematica Package for the Analysis of Quantum Extended Games

Quantum games have attracted much attention in recent years due to their ability to solve decision-making dilemmas. The aim of this study is to extend previous work on quantum games by introducing a Mathematica package QEGS (Quantum Extension Game Solver) dedicated to the study of quantum extensions of classical $2\times2$ games based on the EWL scheme. The package generates all possible game extensions with one or two unitary strategies, which are invariant with respect to isomorphic transformations of the initial games. The package includes a number of functions to study these extensions, such as determining their Nash equilibria in pure strategies, eliminating dominated strategies, or computing maximin strategies. Independently of quantum extensions, these functions can also be used to analyze classical games. Reporting to a pdf is available. The discussion includes an outline of future research directions, such as the exploration of mixed-strategy Nash equilibria and potential real-world applications in fields like quantum computing and secure communications.

quant-ph↗

arXiv2024

Nash equilibria in four-strategy quantum game extensions of the Prisoner's Dilemma

This paper investigates Nash equilibria in pure strategies for quantum approach to the Prisoner's Dilemma. The quantization process involves extending the classical game by introducing two additional unitary strategies. We consider five classes of such quantum games, which remain invariant under isomorphic transformations of the classical game. For each class, we identify and analyse all possible Nash equilibria. Our results reveal the complexity and diversity of strategic behaviour in the quantum setting, providing new insights into the dynamics of classical decision-making dilemmas. In the case of the standard Prisoner's Dilemma, the resulting Nash equilibria of quantum extensions are found to be closer to Pareto optimal solutions than those of the classical equilibrium.

quant-ph↗

arXiv2024

Permissible four-strategy quantum extensions of classical games

The study focuses on strategic-form games extended in the Eisert-Wilkens-Lewenstein scheme by two unitary operations. Conditions are determined under which the pair of unitary operators, along with classical strategies, form a game invariant under isomorphic transformations of the input classical game. These conditions are then applied to determine these operators, resulting in five main classes of games satisfying the isomorphism criterion, and a theorem is proved providing a practical criterion for this isomorphism. The interdependencies between different classes of extensions are identified, including limit cases in which one class transforms into another.

quant-ph↗

arXiv2024

Permissible extensions of classical to quantum games combining three strategies

We study the extension of classical games to the quantum domain, generated by the addition of one unitary strategy to two classical strategies of each player. The conditions that need to be met by unitary operations to ensure that the extended game is invariant with respect to the isomorphic transformations of the input game are determined. It has been shown that there are three types of these extensions, two of them are purely quantum. On the other hand, it has been demonstrated that the extensions of two versions of the same classical game by a unitary operator that does not meet these conditions may result in quantum games that are non-equivalent, e.g. having different Nash equilibria. We use the obtained results to extend the classical Prisoner's Dilemma game to a quantum game that has a unique Nash equilibrium closer to Pareto-optimal solutions than the original one.

quant-ph↗

arXiv2022

Transactional Interpretation for the Principle of Minimum Fisher Information

The principle of minimum Fisher information states that in the set of acceptable probability distributions characterizing the given system, it is best done by the one that minimizes the corresponding Fisher information. This principle can be applied to transaction processes, the dynamics of which can be interpreted as the market tendency to minimize the information revealed about itself. More information involves higher costs (information is physical). The starting point for our considerations is a description of the market derived from the assumption of minimum Fisher information for a strategy with a fixed financial risk. Strategies of this type that minimize Fisher information overlap with the well-known eigenstates of a the quantum harmonic oscillator. The analytical extension of this field of strategy to the complex vector space (traditional for quantum mechanics) suggests the study of the interference of the oscillator eigenstates in terms of their minimization of Fisher information. It is revealed that the minimum value of Fisher information of the superposition of the two strategies being the ground state and the second excited state of the oscillator, has Fisher information less than the ground state of the oscillator. Similarly, less information is obtained for the system of strategies (the oscillator eigenstates) randomized by the Gibbs distribution. We distinguish two different views on the description of Fisher information. One of them, the classical, is based on the value of Fisher information. The second, we call it transactional, expresses Fisher information from the perspective of the constant risk of market strategies. The orders of the market strategies derived from these two descriptions are different. From a market standpoint, minimizing Fisher information is equivalent to minimizing risk.

q-fin.GN↗

arXiv2005

Orbital magnetic moments in pure and doped carbon nanotubes

The unusual band structure of carbon nanotubes (CNs) results in their remarkable magnetic properties. The application of magnetic field parallel to the tube axis can change the conducting properties of the CN from metallic to semiconducting and vice versa. Apart from that B induces (via the Bohm-Aharonov effect) orbital magnetic moments $μ_{orb}$ in the nanotube. These moments are studied both in pure and hole- or electron-doped CNs, isolated or in a circuit. Remarkably, $μ_{orb}$ in pure CNs depends uniquely on their original conducting properties, length, and temperature, but it does not depend on the nanotube radius or the particular chirality. In doped nanotubes the magnetic moments can be strongly altered and depend on the radius and chirality.Temperature can even change their character from diamagnetic at low T to paramagnetic at high T. A full electron-hole symmetry in doped tubes is also revealed.

cond-mat.mes-hall↗

arXiv1996

The Canonical Form of the Rabi Hamiltonian

The Rabi Hamiltonian, describing the coupling of a two-level system to a single quantized boson mode, is studied in the Bargmann-Fock representation. The corresponding system of differential equations is transformed into a canonical form in which all regular singularities between zero and infinity have been removed. The canonical or Birkhoff-transformed equations give rise to a two-dimensional eigenvalue problem, involving the energy and a transformational parameter which affects the coupling strength. The known isolated exact solutions of the Rabi Hamiltonian are found to correspond to the uncoupled form of the canonical system.

quant-ph↗