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IEEE Conference on Decision and Control (CDC)

Constructive control of quantum-mechanical systems

D.R. Sahoo, Murti V. Salapaka

Published Nov 13, 2002• 1 citation
Publisher

Abstract

In this paper we develop a method for exact constructive controllability of quantum mechanical systems. The method has three steps, first a path from the initial state to the final state is determined and intermediate points chosen such that any two consecutive points are close, next small sinusoidal control signals are used to drive the system between the points, and finally quantum measurement technique is used to exactly achieve the desired state. The methodology is demonstrated for the control of spin-half particles in a Stern-Gerlach setting.

DOI
10.1109/.2001.981128

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In this paper we develop a method for exact constructive controllability of quantum mechanical systems. The method has three steps, first a path from the initial state to the final state is determined and intermediate points chosen such that any two consecutive points are close, next small sinusoidal control signals are used to drive the system between the points, and finally quantum measurement technique is used to exactly achieve the desired state. The methodology is demonstrated for the control of spin-half particles in a Stern-Gerlach setting.
In this paper we develop a method for exact constructive controllability of quantum mechanical systems. The method has three steps, first a path from the initial state to the final state is determined and intermediate points chosen such that any two consecutive points are close, next small sinusoidal control signals are used to drive the system between the points, and finally quantum measurement technique is used to exactly achieve the desired state. The methodology is demonstrated for the control of spin-half particles in a Stern-Gerlach setting.
Constructive procedures for the control of a class of coupled spin systems with bounded amplitude sinusoidal pulses are described. Full details are first illustrated for a single spin 1/2 particle. The key steps for a pair of such particles are then sketched. The fact that the amplitudes can be arbitrarily bounded implies that the separation between the Larmor frequencies of the individual spin 1/2 particles can be quite small and yet the system can be controlled by addressing each spin individually.
Constructive procedures for the control of a class of coupled spin systems with bounded amplitude sinusoidal pulses are described. Full details are first illustrated for a single spin 1/2 particle. The key steps for a pair of such particles are then sketched. The fact that the amplitudes can be arbitrarily bounded implies that the separation between the Larmor frequencies of the individual spin 1/2 particles can be quite small and yet the system can be controlled by addressing each spin individually.
A rapid incoherent control scheme is proposed for driving a quantum system from an initial mixed state to a final state having a high fidelity with a target pure state. This scheme consists of two main steps: rapid purification of the initial mixed state and time-optimal control of the conditional state. The first step rapidly purifies the initial mixed state into an almost-pure state (conditional state) through continuous measurements and feedback control. The second step finds a set of suitable time-optimal controls through the control design of a reference model to drive the quantum system from the conditional state to the final state. The switching time between the two steps is determined by the expected purity or fidelity and the resultant state is an almost-pure state having a high fidelity with the target pure state. As an illustrative example, the authors apply the present rapid incoherent control scheme to a spin system. The potential applications of model predictive control to quantum systems are also discussed briefly.
An incoherent control scheme for state control of locally controllable quantum systems is proposed. This scheme includes three steps: (1) amplitude amplification of the initial state by a suitable unitary transformation, (2) projective measurement of the amplified state, and (3) final optimization by a unitary controlled transformation. The first step increases the amplitudes of some desired eigenstates and the corresponding probability of observing these eigenstates, the second step projects, with high probability, the amplified state into a desired eigenstate, and the last step steers this eigenstate into the target state. Within this scheme, two control algorithms are presented for two classes of quantum systems. As an example, the incoherent control scheme is applied to the control of a hydrogen atom by an external field. The results support the suggestion that projective measurements can serve as an effective control and local controllability information can be used to design control laws for quantum systems. Thus, this scheme establishes a subtle connection between control design and controllability analysis of quantum systems and provides an effective engineering approach in controlling quantum systems with partial controllability information.
Quantum nonlocality and quantum steering are fundamental correlations of quantum systems which cannot be created using classical resources only. Nonlocality describes the ability to influence the possible results of measurements carried out in distant systems, in quantum steering where Alice remotely steers Bob's state. Research in nonlocality and steering is of fundamental interest for the development of quantum information and in many applications requiring nonlocal resources like quantum key distribution. On the other hand, the Stern-Gerlach experiment holds an important place in the history, development, and teaching of quantum mechanics and quantum information. In particular, the thought experiment of consecutive Stern-Gerlach experiments is commonly used to exemplify the concept of noncommutativity between quantum operators. However, to the best of our knowledge, the consecutive Stern-Gerlach experiments have not been treated in a fully fledged quantum manner yet, and it is a widely accepted idea that atoms crossing consecutive Stern-Gerlach experiments follow classical paths. Here we demonstrate that two consecutive Stern-Gerlach experiments generate nonlocality and steering, and these nonlocal effects strongly modify our usual understanding of this experiment. Also, we discuss the implications of this result and its relation with entanglement. This suggests the use of quantum correlations, of particles possessing mass, to generate nonlocal tasks using this experiment.