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Chinese Physics

Enhancing the capability of controlling quantum systems via ancillary systems

Zhang Ming, Gao Da-Yuan, Dai Hong-Yi, Xie Hong-Wei, 胡德文

Published Mar 20, 2007• 0 citations
Publisher

Abstract

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.

DOI
10.1088/1009-1963/16/4/005

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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.
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.
We demonstrate how structured decompositions of unitary operators can be employed to derive control schemes for finite-level quantum systems that require only sequences of simple control pulses such as square wave pulses with finite rise and decay times or Gaussian wavepackets. To illustrate the technique, it is applied to find control schemes to achieve population transfers for pure-state systems, complete inversions of the ensemble populations for mixed-state systems, create arbitrary superposition states and optimize the ensemble average of dynamic observables.
This paper presents a constructive proof of complete kinematic state controllability of finite-dimensional open quantum systems whose dynamics are represented by Kraus maps. For any pair of states (pure or mixed) on the Hilbert space of the system, we explicitly show how to construct a Kraus map that transforms one state into another. Moreover, we prove by construction the existence of a Kraus map that transforms all initial states into a predefined target state (such a process may be used, for example, in quantum information dilution). Thus, in sharp contrast to unitary control, Kraus-map dynamics allows for the design of controls which are robust to variations in the initial state of the system. The capabilities of non-unitary control for population transfer between pure states illustrated for an example of a two-level system by constructing a family of non-unitary Kraus maps to transform one pure state into another. The problem of dynamic state controllability of open quantum systems (i.e., controllability of state-to-state transformations, given a set of available dynamical resources such as coherent controls, incoherent interactions with the environment, and measurements) is also discussed.
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.
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.
In this thesis, we study the controllability and accessibility properties of quantum systems. First, we provide a new proof of the fact that, in general, open quantum systems are not controllable. By controllability we mean that for every pair of initial and target states of the system there exists a finite time T>0 and a control u such that by adding u into the system’s evolution the target state can be reached from the initial state in time T. We then consider a bipartite open system A+B consisting of two interacting qubits A and B under the assumption that only A is directly affected by control u and only A interacts with the environment. We investigate the controllability properties of B. We show that target pure states of B are not reachable in finite time from initial mixed states of B when the initial state of A+B lies in the interior of the set of states. We then examine whether it is possible to maintain the state of B pure. For three specific couplings between A and B we find that the only admissible trajectories are either unaffected by the control u or constant. These results constitute a first negative answer regarding the indirect manipulation of B. Finally, we study the accessibility of B, that is, whether for every initial state of B the attainable set contains an open subset of the set of states of B. For this property, we provide a sufficient condition that can also be applied to more general settings, such as those arising in geometric control theory.