Kinematics: How Quantum Systems are Described

Pavel Exner

Re-Bing Wu, Constantin Brif, M. R. James +1 more

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 where 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.

Mario Schönberg

J.V. Leahy, S. G. Schirmer

Girardeau et al. (1998) derived kinematical bounds on the optimization of observables for mixed-state quantum systems and showed that they are dynamically realizable if the system is completely controllable. In this paper the problem of finding dynamically realizable bounds for systems that are not completely controllable is addressed. We derive such bounds for systems whose dynamics can be decomposed into subspace dynamics. We also study systems that are not decomposable yet fail to be completely controllable. For these systems, the question of dynamical realizability of the kinematical bounds depends on the accessibility of the target states for which the expectation value of the observable assumes its kinematical maximum.