Towards the theory of control in observable quantum systems

TL;DR

This paper develops a theoretical framework for controlling observable quantum systems using quantum filtering, sufficient coordinates, and Bellman equations, with applications to systems observed at discrete times.

Contribution

It introduces a formalism for quantum control based on quantum filtering and sufficient coordinates, and derives Bellman equations for optimal control in this context.

Findings

Quantum filtering describes state reduction during measurements.

Sufficient coordinates form a classical Markov process.

Bellman equation for quantum control is derived.

Abstract

An operational description of the controlled Markov dynamics of quantum-mechanical system is introduced. The feedback control strategies with regard to the dynamical reduction of quantum states in the course of quantum real-time measurements are discribed in terms of quantum filtering of these states. The concept of sufficient coordinates for the description of the a posteriori quantum states from a given class is introduced, and it is proved that they form a classical Markov process with values in either state operators or state vector space. The general problem of optimal control of a quantum-mechanical system is discussed and the corresponding Bellman equation in the space of sufficient coordinates is derived. The results are illustrated in the example of control of the semigroup dynamics of a quantum system that is instantaneously observed at discrete times and evolves between measurement times according to the Schroedinger equation.