Loss of Memory and Convergence of Quantum Markov Processes

TL;DR

This paper studies the behavior of quantum Markov processes, showing that information about the initial state diminishes over time and converges, with a new framework for understanding information loss and ergodicity.

Contribution

It introduces an equivalence class framework for quantum states to analyze information loss and characterizes weak ergodicity in quantum Markov processes.

Findings

Information about initial states is monotone non-increasing and converges.

The trajectory of quantum states converges to a point in the space of equivalence classes.

Weak ergodicity is characterized within this new framework.

Abstract

In a quantum (inhomogeneous) Markov process $\rho_1:=\Gamma_1(\rho)$, $\rho_2:=\Gamma_1(\rho_1)$, ..., where $\Gamma_i$ are CPTP maps and $\rho$ is the initial state, the the state of the system is either oscillatory or convergent to a point or convergent to an oscillatory orbit. Whichever the case it is, "information" about the initial state is always monotone non-increasing and convergent. This fact motivate us to define an equivalence class of families of quantum states, which embodies the bundle of all "information quantities" about the initial state. We show, for any quantum inhomogeneous Markov process over a finite dimensional Hilbert space, the trajectory in the space of the all equivalence classes is "monotone decreasing" and convergent to a point, relative to a reasonablly defined topology. Also, a characterization of weak ergodicity in this picture is given.