Jaynes principle for quantum Markov processes: Generalized Gibbs - von Neumann states rule

TL;DR

This paper establishes a generalized Jaynes principle for quantum Markov processes, showing that asymptotic states are Gibbs-like and derived from quantum relative entropy, extending statistical physics concepts.

Contribution

It introduces a new formulation of the Jaynes principle for quantum Markov processes based on quantum relative entropy, applicable in various settings with different prior knowledge.

Findings

Asymptotic states are Gibbs-like and derived from quantum relative entropy.

The principle applies to both discrete and continuous quantum Markov processes.

It generalizes the MaxEnt principle and modifies fundamental statistical physics relations.

Abstract

We prove that any asymptotics of a finite-dimensional quantum Markov processes can be formulated in the form of a generalized Jaynes principle in the discrete as well as in the continuous case. Surprisingly, we find that the open system dynamics does not require maximization of von Neumannentropy. In fact, the natural functional to be extremized is the quantum relative entropy and the resulting asymptotic states or trajectories are always of the exponential Gibbs-like form. Three versions of the principle are presented for different settings, each treating different prior knowledge: for asymptotic trajectories of fully known initial states, for asymptotic trajectories incompletely determined by known expectation values of some constants of motion and for stationary states incompletely determined by expectation values of some integrals of motion. All versions are based on the knowledge of the underlying dynamics. Hence our principle is primarily rooted in the inherent physics and it is not solely an information construct. The found principle coincides with the MaxEnt principle in the special case of unital quantum Markov processes. We discuss how the generalized principle modifies fundamental relations of statistical physics.