An ergodic theorem for quantum processes with applications to matrix product states

TL;DR

This paper establishes an ergodic theorem for quantum processes modeled by sequences of quantum channels, demonstrating exponential convergence to a steady state and analyzing correlation decay in matrix product states.

Contribution

It introduces a new ergodic theorem for quantum processes driven by ergodic dynamical systems, with applications to matrix product states.

Findings

Quantum states converge exponentially fast to an ergodic state.

2-point correlations in matrix product states decay exponentially.

The thermodynamic limit of ergodic matrix product states is characterized.

Abstract

Any discrete quantum process is represented by a sequence of quantum channels. We consider ergodic quantum processes obtained by a map that takes the points along the trajectory of a discrete ergodic dynamical system to the space of quantum channels. Under natural irreducibility conditions, we obtain a theorem showing that the state under such a process converges exponentially fast to an ergodic sequence depending on the process, but independent of the initial state. As an application, we describe the thermodynamic limit of ergodic matrix product states and prove that the 2-point correlations of local observables in such states decay exponentially with their distance in the bulk.