Heisenberg Picture for Open Quantum Systems

TL;DR

This paper develops a comprehensive Heisenberg picture framework for open quantum systems, introducing multiple image operators per observable and deriving perturbative expressions that account for system-environment interactions.

Contribution

It presents a complete formulation of the Heisenberg picture for open quantum systems, addressing previous incompleteness and deriving perturbative expressions for image operators.

Findings

Multiple image operators are needed for each observable.

Perturbative expressions depend non-linearly on environment state.

In the Markovian limit, operators follow an adjoint Lindblad equation.

Abstract

In this note, we develop a framework to describe open quantum systems in the Heisenberg picture, i.e., via time evolving operator algebras. We point out the incompleteness of the previous proposals in this regard. We argue that a complete Heisenberg picture for an open quantum system involves multiple image Heisenberg operators for each system observable. For a given system observable, the number of such image operators is equal to the dimension of the environment Hilbert space. We derive a perturbative expression, accurate upto arbitrary orders in the system environment coupling, for these image operators in terms of a single one point operator. This expression depends non-linearly on the state of the environment. This perturbative expression can equivalently be thought of as deforming the operator product on the Hilbert space of the open quantum system. In the Markovian limit, the one point operator evolves by an adjoint Lindblad equation. We illustrate these ideas using a simple spin system.