Density Matrix RNN (DM-RNN): A Quantum Information Theoretic Framework for Modeling Musical Context and Polyphony

TL;DR

The paper introduces the Density Matrix RNN (DM-RNN), a novel quantum-inspired neural architecture that models musical context and polyphony by capturing ambiguity and entanglement using quantum information theory.

Contribution

It presents a new RNN architecture based on density matrices, enabling the modeling of musical ambiguity and coherence with a rigorous quantum information framework.

Findings

Captures musical ambiguity using mixed states.

Quantifies uncertainty with Von Neumann Entropy.

Measures entanglement between voices with Quantum Mutual Information.

Abstract

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.