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Proceedings of the ... American Control Conference/Proceedings of the American Control Conference

Generalized Controllability of Quantum Systems: Beyond the Limitation of Open-loop Controllability

Ming Zhang, Dewen Hu, Hong‐Yi Dai, Hongwei Xie

Published Jul 1, 2007• 0 citations
Publisher

Abstract

To control a quantum system, not only may an ancillary system be introduced, but unitary operations and quantum projective measurement can also be performed on the quantum system or the resulting composite system. As a quantum system may be "physically" controllable even if the quantum system is not open-loop controllable, it is, therefore, necessary to go beyond the limitation of open-loop controllability for quantum systems. By studying the link between quantum generalized measurement and generalized pure state reachability, we further clarify the rationality of redefining the concept of generalized controllability from a view point of physical operation. It is also demonstrated that applying unitary operations, performing quantum projective measurement and introducing ancillary systems can play important roles in controlling quantum systems. Therefore, instead of adopting the concept of open-loop controllability for quantum systems, we need to define and discuss the concept of generalized controllability from a view point of physical operation. This is in marked contrast to the fact that the "physical" controllability for classical systems can only be discussed in the sense of open- loop control.

DOI
10.1109/acc.2007.4282257

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To control a quantum system, not only may an ancillary system be introduced, but unitary operations and quantum projective measurement can also be performed on the quantum system or the resulting composite system. As a quantum system may be "physically" controllable even if the quantum system is not open-loop controllable, it is, therefore, necessary to go beyond the limitation of open-loop controllability for quantum systems. By studying the link between quantum generalized measurement and generalized pure state reachability, we further clarify the rationality of redefining the concept of generalized controllability from a view point of physical operation. It is also demonstrated that applying unitary operations, performing quantum projective measurement and introducing ancillary systems can play important roles in controlling quantum systems. Therefore, instead of adopting the concept of open-loop controllability for quantum systems, we need to define and discuss the concept of generalized controllability from a view point of physical operation. This is in marked contrast to the fact that the "physical" controllability for classical systems can only be discussed in the sense of open- loop control.
This paper explores the potential of controlling quantum systems by introducing ancillary systems and then performing unitary operation on the resulting composite systems. It generalizes the concept of pure state controllability for quantum systems and establishes the link between the operator controllability of the composite system and the generalized pure state controllability of its subsystem. It is constructively demonstrated that if a composite quantum system can be transferred between any pair of orthonormal pure vectors, then its subsystem is generalized pure-state controllable. Furthermore, the unitary operation and the coherent control can be concretely given to transfer the system from an initial state to the target state. Therefore, these properties may be potentially applied in quantum information, such as manipulating multiple quantum bits and creating entangled pure states. A concrete example has been given to illustrate that a maximally entangled pure state of a quantum system can be generated by introducing an ancillary system and performing open-loop coherent control on the resulting composite system.
For any specified pure state of quantum open system, we can construct a kind of quantum generalized measurement (QGM) that the state of the system after measurement will be deterministically collapsed into the specified pure state from any initial state. In other words, any pure state of quantum open system is reachable by QGM. Subsequently, whether the qubit is density matrix controllable is discussed in the case of pure dephasing. Our results reveal that combining QGM with coherent control will enhance the ability of controlling the quantum open system. Furthermore, it is found that the ability to perform QGM on the quantum open system, combined with the ability of coherence control and conditions of decoherence-free subspace, allows us to suppress quantum decoherence.
The paper first briefly introduces the current research on quantum control. Then the quantum open-loop control (no measurement introduced) and the closed-loop control (measurement introduced) was compared through a standard model of coherent control. When using quantum closed-loop control, we need measurement which inevitably brings in the reduction noise. This is different from the classical control systems. On the other hand, measurement-based feedback can reduce the influence of the system uncertainties. Thus, there is a tradeoff between the noise influence brought in by the measurement and the information got by the measurement. The paper demonstrate that in the case without decoherence, the quantum closed-loop control is superior to the open-loop control in dealing with the uncertainty in some sense. In the case with decoherence, we discuss the controllability of the quantum control through two performance indices.
Quantum phenomena of interest in connection with computation and communication often deal with transfers between eigenstates. For systems having only a finite number of states, the quantum evolution equation is finite-dimensional and the results on controllability on Lie groups as worked out decades ago provide most of what is needed in so far as controllability of dissipationless systems is concerned. However, for infinite-dimensional evolution of quantum systems, many difficulties, both conceptual and technical, remain. In this paper we organize some recent results from the physics literature in control theoretic terms and emphasize the type of analysis needed to go beyond what basic differential geometry can provide. In particular, we analyze from a controllability point of view the important results of Law and Eberly, and discuss the problem of controllability of quantum systems subject to the constraint that the trajectories must lie in pre-defined subspaces. The study of this problem is motivated by recent work on the design of quantum computers that avoid loss of coherence by operating in decoherence free subspaces.
Feedback plays an essential role in classical control theory, which has been investigated in depth. However, in quantum control theory, the study of feedback is still in its infancy. The paper continues to explore the fundamental characteristics of quantum feedback control for a typical coherent model based on the results of [3] by highlighting the significant differences between quantum feedback control and classical feedback control due to quantum measurement. The paper compares the control performance of quantum feedback control and quantum open-loop control in the cases with and without environment-induced decoherence. It concludes that in some sense, the quantum feedback control is still superior to the quantum open-loop control despite the complicated tradeoff in the feedback case, and further points out that how the quantum measurement-induced-collapse can play a positive role in the quantum feedback control.
This paper demonstrates that the decoherence-free subsystem of open quantum dynamical system is open-loop controllable if the suitable coherent controls are permitted. This result is in marked contrast to the fact that open quantum dynamical systems are not open-loop controllable. In other words, combining the decoherence-free subsystem condition with coherent control on the open quantum dynamical systems will enhance the ability to suppress the decoherence and to manipulate open quantum dynamical systems.