Deterministic Equations for Feedback Control of Open Quantum Systems II: Properties of the memory function

TL;DR

This paper develops a theoretical framework to analyze the properties of memory functions in feedback control of open quantum systems, linking classical memory dynamics with quantum monitoring and applying it to stabilize quantum states.

Contribution

It introduces a hybrid bipartite state framework and information-theoretic measures to characterize quantum feedback memory effects, a novel approach in quantum control theory.

Findings

Memory can be modeled as a classical system coupled to quantum systems.

The framework quantifies system-memory correlations using information theory.

Application to a two-level system demonstrates stabilization of quantum states.

Abstract

Feedback uses past detection outcomes to dynamically modify a quantum system and is central to quantum control. These outcomes can be stored in a memory, defined as a stochastic function of past measurements. In this work, we investigate the main properties of a general memory function subject to arbitrary feedback dynamics. We show that the memory can be treated as a classical system coupled to the monitored quantum system, and that their joint evolution is described by a hybrid bipartite state. This framework allows us to introduce information-theoretic measures that quantify the correlations between the system and the memory. Furthermore, we develop a general framework to characterize the statistics of the memory -- such as moments, cumulants, and correlation functions -- which can be applied both to general feedback-control protocols and to monitored systems without feedback. As an application, we analyze feedback schemes based on detection events in a two-level system coupled to a thermal bath, focusing on protocols that stabilize either the excited-state population or Rabi oscillations against thermal dissipation.