Loss of Memory and Convergence of Quantum Markov Processes

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.