Dynamics of Matrix Product States in the Heisenberg Picture: Projectivity, Ergodicity, and Mixing
Abdessatar Souissi, Amenallah Andolsi
TL;DR
This work develops a Heisenberg-picture framework for Matrix Product States (MPS) and infinite MPS (iMPS), enabling rigorous analysis of their correlation structure, ergodicity, and thermodynamic limits. By distinguishing projective MPS from non-projective ones and employing a quantum Markov-Dobrushin criteria, the authors establish conditions under which infinite-volume limits exist and classify MPS as ergodic or mixing. They provide a rigorous construction of infinite-volume states through matrix-product-operator sequences, with a GHZ-state example and a depolarizing-channel case study illustrating both finite- and infinite-range behaviors. The results deepen the algebraic understanding of MPS/iMPS, linking them to quantum Markov chains and finitely correlated states and shedding light on entanglement, phase structure, and dynamical properties in quantum spin chains.
Abstract
This paper introduces a Heisenberg picture approach to Matrix Product States (MPS), offering a rigorous yet intuitive framework to explore their structure and classification. MPS efficiently represent ground states of quantum many-body systems, with infinite MPS (iMPS) capturing long-range correlations and thermodynamic behavior. We classify MPS into projective and non-projective types, distinguishing those with finite correlation structures from those requiring ergodic quantum channels to define a meaningful limit. Using the Markov-Dobrushin inequality, we establish conditions for infinite-volume states and introduce ergodic and mixing MPS. As an application, we analyze the depolarizing MPS, highlighting its lack of finite correlations and the need for an alternative ergodic description. This work deepens the mathematical foundations of MPS and iMPS, providing new insights into entanglement, phase transitions, and quantum dynamics.