An Exponential Mixing Condition for Quantum Channels: Application to Matrix Product States

Abstract

Quantum channels are fundamental tools in quantum information processing, enabling state transformations within quantum systems, secure communication, and error correction. Understanding their ergodic and mixing properties is crucial for characterizing their long-term behavior. In this paper, we establish a sufficient condition for mixing via a quantum Markov–Dobrushin inequality, demonstrating that quantum channels with a positive Markov–Dobrushin constant exhibit exponential convergence to their stationary state. Furthermore, we present a theorem linking mixing quantum channels to the thermodynamic limit of matrix product states (MPS), providing a rigorous foundation for understanding the stability and ergodicity of MPS in infinite quantum systems. To illustrate the applicability of our results, we analyze the qubit depolarizing channel, showcasing its mixing behavior and implications for quantum information tasks.