Dynamical programming of continuously observed quantum systems

TL;DR

This paper develops dynamical programming methods for optimal control of quantum systems under continuous observation, deriving generalized Hamilton-Jacobi-Bellman equations and exploring strategies for state purification.

Contribution

It introduces a unified framework for quantum optimal control with continuous measurements, including feedback and open loop schemes, and derives new equations for control under observation constraints.

Findings

Derived a generalized Hamilton-Jacobi-Bellman equation for quantum feedback control.

Demonstrated the formalism with a controlled qubit example.

Discussed optimal observation strategies for state purification.

Abstract

We develop dynamical programming methods for the purpose of optimal control of quantum states with convex constraints and concave cost and bequest functions of the quantum state. We consider both open loop and feedback control schemes, which correspond respectively to deterministic and stochastic Master Equation dynamics. For the quantum feedback control scheme with continuous non-demolition observations we exploit the separation theorem of filtering and control aspects for quantum stochastic dynamics to derive a generalized Hamilton-Jacobi-Bellman equation. If the control is restricted to only Hamiltonian terms this is equivalent to a Hamilton-Jacobi equation with an extra linear dissipative term. In this work, we consider, in particular, the case when control is restricted to only observation. A controlled qubit is considered as an example throughout the development of the formalism. Finally, we discuss optimum observation strategies to obtain a pure state from a mixed state of a quantum two-level system.