Conditional expectations associated with quantum states

TL;DR

This paper extends classical conditional expectations to the quantum realm within Jordan operator algebras, establishing conditions for their existence, and explores quantum Markov processes and their generators.

Contribution

It introduces a weak compatibility criterion for quantum conditional expectations and extends classical Markov process concepts to quantum systems.

Findings

Quantum conditional expectations exist only under a specific compatibility criterion.

A framework for quantum Markov processes is developed within Jordan operator algebras.

Dynamical semigroups and their generators are characterized for quantum observables.

Abstract

An extension of the conditional expectations (those under a given subalgebra of events and not the simple ones under a single event) from the classical to the quantum case is presented. In the classical case, the conditional expectations always exist; in the quantum case, however, they exist only if a certain weak compatibility criterion is satisfied. This compatibility criterion was introduced among others in a recent paper by the author. Then, state-independent conditional expectations and quantum Markov processes are studied. A classical Markov process is a probability measure, together with a system of random variables, satisfying the Markov property and can equivalently be described by a system of Markovian kernels (often forming a semigroup). This equivalence is partly extended to quantum probabilities. It is shown that a dynamical (semi)group can be derived from a given system of quantum observables satisfying the Markov property, and the group generators are studied. The results are presented in the framework of Jordan operator algebras, and a very general type of observables (including the usual real-valued observables or self-adjoint operators) is considered.