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

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.