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.

翻译：经典循环神经网络（RNN）将音乐上下文压缩为一个确定性的隐藏状态向量，这施加了一个信息瓶颈，无法捕捉音乐中固有的模糊性。我们提出密度矩阵循环神经网络（DM-RNN），这是一种利用密度矩阵的新型理论架构。该模型能够维持一个音乐解释的统计系综（混合态），同时捕捉经典概率与量子相干性。我们使用量子信道（CPTP映射）严格定义了其时间动力学。关键的是，我们详细阐述了一种基于Choi-Jamiolkowski同构的参数化策略，通过构造确保学习到的动力学始终保持物理有效性（CPTP）。我们引入了一个使用冯·诺依曼熵来量化音乐不确定性的分析框架，以及使用量子互信息（QMI）来度量声部间纠缠的分析方法。DM-RNN为建模复杂、模糊的音乐结构提供了一个数学上严谨的框架。