Abstract We present a time-dependent quantum game-theoretic framework to analyze general two-voice contrapuntal structures. Utilizing the polyphony of J.S. Bach as a representative proof of concept, we model the functional interdependence of independent voices using a multi-contextual binary abstraction of tonal harmony. By incorporating an entanglement parameter and a payoff penalty reflecting asymmetric musical roles, we extract underlying quantum strategies from symbolic musical data. Our optimization demonstrates that the game-theoretic model assuming quantum entanglement reproduces empirical probability distributions with significantly higher accuracy than the corresponding unentangled baseline model. Furthermore, applying the payoff penalty drives the system beyond classical limits, revealing that the structural demands of counterpoint actively exploit nonlocal quantum correlations. Finally, we show that the resulting payoff-induced probabilistic deviations can be mathematically mapped to temporal timing fluctuations. This exploratory study provides a physics-based foundation for interpreting structurally driven deviations from a musical score as emergent temporal dynamics in creative performance.
Abstract We present a time-dependent quantum game-theoretic framework to analyze general two-voice contrapuntal structures. Utilizing the polyphony of J.S. Bach as a representative proof of concept, we model the functional interdependence of independent voices using a multi-contextual binary abstraction of tonal harmony. By incorporating an entanglement parameter and a payoff penalty reflecting asymmetric musical roles, we extract underlying quantum strategies from symbolic musical data. Our optimization demonstrates that the game-theoretic model assuming quantum entanglement reproduces empirical probability distributions with significantly higher accuracy than the corresponding unentangled baseline model. Furthermore, applying the payoff penalty drives the system beyond classical limits, revealing that the structural demands of counterpoint actively exploit nonlocal quantum correlations. Finally, we show that the resulting payoff-induced probabilistic deviations can be mathematically mapped to temporal timing fluctuations. This exploratory study provides a physics-based foundation for interpreting structurally driven deviations from a musical score as emergent temporal dynamics in creative performance.
Abstract This article reports on a series of mathe-musical experiments based on the dynamic behavior of a musical quantum model of C-major tonality. The model reinterprets empirical pitch-class profiles of tonal attraction in C-major as the probability wave function of a one-dimensional stationary ground state within a perturbed quantum harmonic oscillator. By solving the associated time-dependent Schrödinger equation, we assign specific temporal evolutions to any given initial state. We explore two distinct sonification strategies: First, the time-evolutions of various initial states—in particular deflections of the ground state and higher excited states—are used to drive a stroboscopic stratification of musical hierarchies and to set them into Minimal Music-inspired travelling note patterns. Secondly, we implement rich continuous sound processes that rely on the uninterrupted flow of the wave function. Crucially, these approaches bypass the simulation of quantum measurements; instead, they leverage the full theoretical information of the wave function’s evolution.
We study how strategic interaction can arise from controlled quantum dynamics rather than being imposed as an external mathematical structure. We introduce a class of interaction-defined quantum games in which players are represented by distinguishable quantum walkers, strategies correspond to local coin operations, and payoffs are defined as expectation values of physical observables. Using interacting discrete-time quantum walks as a concrete platform, we demonstrate numerically that competitive, cooperative, and asymmetric games admit stable stationary strategy profiles when the walkers are coupled, while no non-trivial equilibria exist in the absence of interaction. To clarify the game-theoretic structure, we derive an analytic perturbative decomposition of the payoff function in the weak-interaction regime, showing explicitly that strategic coupling originates from interaction-induced interference terms in the joint probability distribution. For a collision-based phase interaction, the payoff becomes non-separable at first order in the interaction strength and generically admits stationary points satisfying the Nash conditions. Our results provide a physically explicit realization of strategic interdependence in quantum transport processes and establish interacting quantum walks as a minimal platform for studying game-theoretic behavior emerging from unitary dynamics.
We present a web-based interactive music system that generates sound through the framework of twoplayer quantum game theory. Two users connect via a internet browser interface, select quantum strategies, and trigger notes by pressing piano keys. Each input initiates a two-qubit circuit whose measurement probabilistically collapses into one of four outcomes, each linked to a payoff pair inspired by the Prisoner’s Dilemma. These sampled payoffs determine pitch shifts relative to a neutral baseline, producing distinct sounds for each player.The system provides real-time visual and auditory feedback, allowing participants to perceive how their own strategies intertwine with those of their opponent through quantum interference and measurement. Beyond applications in music and education, the framework also models improvisatory interplay-akin to jazz performance-where each player’s decision reshapes the shared horizon of possible outcomes. This platform opens new avenues for exploring quantum games, human-computer interaction, and entangled decision-making through sound.
Classical Recurrent Neural Networks (RNNs) summarize musical context into a deterministic hidden state vector, imposing an information bottleneck that fails to capture the inherent ambiguity in music. We propose the Density Matrix RNN (DM-RNN), a novel theoretical architecture utilizing the Density Matrix. This allows the model to maintain a statistical ensemble of musical interpretations (a mixed state), capturing both classical probabilities and quantum coherences. We rigorously define the temporal dynamics using Quantum Channels (CPTP maps). Crucially, we detail a parameterization strategy based on the Choi-Jamiolkowski isomorphism, ensuring the learned dynamics remain physically valid (CPTP) by construction. We introduce an analytical framework using Von Neumann Entropy to quantify musical uncertainty and Quantum Mutual Information (QMI) to measure entanglement between voices. The DM-RNN provides a mathematically rigorous framework for modeling complex, ambiguous musical structures.
Abstract Multiplayer quantum games have received growing attention due to the ability of quantum computational models to access enlarged strategic spaces, thereby enabling the emergence of nonclassical Nash equilibria and Pareto-efficient solutions. Concurrently, graph-theoretic representations have become a standard tool for encoding relational dependencies among interacting players. This work establishes a unified formalism that embeds arbitrary graph topologies into parametrized quantum-game circuits, providing a direct mapping between network structure and the strategic interaction space of quantum players. Within this framework, we analytically and computationally evaluate how graph-induced interpersonal couplings modulate payoff distributions over repeated quantum-game iterations. In addition, we integrate a reward-driven adaptation algorithm that allows players to optimize their local strategy parameters dynamically with respect to accumulated payoffs. Experimental results demonstrate that the equilibrium reward landscape is highly sensitive to the players’ graph-theoretic centrality and the weighted structure of their adversarial and cooperative relationships.
Quantum games extend classical game theory into the quantum realm by incorporating quantum players and quantum strategies. Two-player quantum games have been widely studied and applied, among other areas, in optimization and decision-making. In this work, we introduce, formalize and analyze a novel class of three-player quantum games, referred to as Proxy Quantum Games (PQGs). In this setting, one player, the Controller, influences or governs the strategic operations executed by a second player, the Proxy, who, in turn, competes directly against a third player, the Opponent. The strategic influence is implemented via controlled unitary operations, whereby the Proxy’s available strategies are conditioned on the Controller’s quantum state. We further generalize the PQG framework by incorporating multiple Controllers acting on a common Proxy, thereby enabling the study of higher-order strategic interactions and multi-agent influence structures. The objective under consideration is the maximization of the Controller’s expected payoff, under various configurations of strategic control and entanglement. Comparative analyses are conducted between the PQG formulation and conventional multiplayer quantum games, highlighting key differences in strategic behavior and payoff distributions.