Limits of optimal control yields achievable with quantum controllers

TL;DR

This paper investigates the fundamental limits of quantum control yields, establishing conditions under which quantum controllers can surpass classical bounds and reach ultimate physical limits in quantum systems.

Contribution

It introduces a comprehensive framework for understanding when quantum controllers can exceed classical bounds and attain the maximum possible control objective in quantum systems.

Findings

Quantum controllers can surpass classical kinematic bounds under certain conditions.

Necessary and sufficient conditions for reaching the quantum kinematic bound are identified.

Examples demonstrate the theoretical conditions with systems in thermal states.

Abstract

In quantum optimal control theory, kinematic bounds are the minimum and maximum values of the control objective achievable for any physically realizable system dynamics. For a given initial state of the system, these bounds depend on the nature and state of the controller. We consider a general situation where the controlled quantum system is coupled to both an external classical field (referred to as a classical controller) and an auxiliary quantum system (referred to as a quantum controller). In this general situation, the kinematic bound is between the classical kinematic bound (CKB), corresponding to the case when only the classical controller is available, and the quantum kinematic bound (QKB), corresponding to the ultimate physical limit of the objective's value. Specifically, when the control objective is the expectation value of a quantum observable (a Hermitian operator on the system's Hilbert space), the QKBs are the minimum and maximum eigenvalues of this operator. We present, both qualitatively and quantitatively, the necessary and sufficient conditions for surpassing the CKB and reaching the QKB, through the use of a quantum controller. The general conditions are illustrated by examples in which the system and controller are initially in thermal states. The obtained results provide a basis for the design of quantum controllers capable of maximizing the control yield and reaching the ultimate physical limit.