Adaptive Quantum Channels as Long-Memory Generative Models

We introduce adaptive quantum channels as a history-dependent extension of completely positive trace-preserving (CPTP) maps. At each step, these channels apply measurement bases conditioned on the history of the observation process. Because the measurement basis directly influences the latent state, the quantum dynamics becomes intrinsically non-Markovian, with temporal correlations between quantum states emerging as entanglement in time. The measurement basis is controlled by an adaptive mechanism that learns short- and long-term, linear and nonlinear dependencies in stochastic processes. To demonstrate the emergence of entanglement in time, we define space-time density operators capturing correlations between evolving quantum states. The inductive bias introduced by the adaptive control is quantified by quantum time-dependency measures. The proposed framework is applied to learn a generative model of mid-price dynamics in a limit order book (LOB). The trained model accurately reproduces both the marginal and joint distributions of mid-price movements and volatility regimes, capturing the complex temporal dependencies characteristic of high-frequency financial data. Finally, we analyze how the history length and the adaptation strategy influence model performance.