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arXiv:1412.5287 · Physical Review A

Limits of optimal control yields achievable with quantum controllers

Re-Bing Wu, Constantin Brif, M. R. James, HERSCHEL A. RABITZ

Published Apr 23, 2015• 11 citations

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 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.

DOI: 10.1103/physreva.91.042327

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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.

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.

Upper and lower bounds are established for time-dependent ensemble averages of observables of driven quantum systems in mixed states. They limit controllability of observables independently of the control fields. Narrower bounds are established when the observable is a projector onto a pure quantum state or subspace. They are optimal in the sense of being kinematically achievable. Calculations on nonlinear optimal control of a four-level model indicate that these kinematical bounds are dynamically achievable asymptotically with increasing control pulse strength.

In this paper we investigate the limits of control for mixed-state quantum systems. The constraint of unitary evolution for nondissipative quantum systems imposes kinematical bounds on the optimization of arbitrary observables. We summarize our previous results on kinematical bounds and show that these bounds are dynamically realizable for completely controllable systems. Moreover, we establish improved bounds for certain partially controllable systems. Finally, the question of dynamical realizability of the bounds for arbitrary partially controllable systems is shown to depend on the accessible sets of the associated control system on the unitary group

U(N)

and the results of a few control computations are discussed briefly.

In this paper we employ optimal control theory to design an external driving field which reduces the difference between certain quantum observables and their classical counterparts in addition to achieving some dynamical objective. Because many different driving fields can achieve a similar target, there is the flexibility of asking that the field also cause quantum and classical observables to match. Several examples are shown using a one-dimensional model system which is highly nonclassical in the absence of a driving field. These illustrations demonstrate some of the richness of possibilities for obtaining good quantum-classical agreement for chosen observables even in the presence of other design criteria.

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.

This paper analyzes classical and quantum physical systems from an optimal control perspective. Specifically, we explore whether their associated dynamics can correspond to an open- or closed-loop feedback evolution of a control problem. Firstly, for the classical regime, when it is viewed in terms of the theory of canonical transformations, we find that a closed-loop feedback problem can describe it. Secondly, for a quantum physical system, if one realizes that the Heisenberg commutation relations themselves can be considered constraints in a non-commutative space, then the momentum must depend on the position of any generic wave function. That implies the existence of a closed-loop strategy for the quantum case. Thus, closed-loop feedback is a natural phenomenon in the physical world. By way of completeness, we briefly review control theory and the classical mechanics of constrained systems and analyze some examples at the classical and quantum levels.