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arXiv:1412.4146 · Physical Review A

Approximation of reachable sets for coherently controlled open quantum systems: Application to quantum state engineering

Jun Li, Dawei Lu, Zhihuang Luo, Raymond Laflamme, Xinhua Peng, Jiangfeng Du

Published Jul 7, 2016• 18 citations
arXivHTMLPDF

Abstract

Precisely characterizing and controlling realistic quantum systems under noises is a challenging frontier in quantum sciences and technologies. In developing reliable controls for open quantum systems, one is often confronted with the problem of the lack of knowledge on the system controllability. The purpose of this paper is to give a numerical approach to this problem, that is, to approximately compute the reachable set of states for coherently controlled quantum Markovian systems. The approximation consists of setting both upper and lower bounds for system's reachable region of states. Furthermore, we apply our reachability analysis to the control of the relaxation dynamics of a two-qubit nuclear magnetic resonance spin system. We implement some experimental tasks of quantum state engineering in this open system at a near optimal performance in view of purity: e.g., increasing polarization and preparing pseudopure states. These results demonstrate the usefulness of our theory and show interesting and promising applications of environment-assisted quantum dynamics.

DOI
10.1103/physreva.94.012312

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Precisely characterizing and controlling realistic quantum systems under noises is a challenging frontier in quantum sciences and technologies. In developing reliable controls for open quantum systems, one is often confronted with the problem of the lack of knowledge on the system controllability. The purpose of this paper is to give a numerical approach to this problem, that is, to approximately compute the reachable set of states for coherently controlled quantum Markovian systems. The approximation consists of setting both upper and lower bounds for system's reachable region of states. Furthermore, we apply our reachability analysis to the control of the relaxation dynamics of a two-qubit nuclear magnetic resonance spin system. We implement some experimental tasks of quantum state engineering in this open system at a near optimal performance in view of purity: e.g., increasing polarization and preparing pseudopure states. These results demonstrate the usefulness of our theory and show interesting and promising applications of environment-assisted quantum dynamics.
We study reachable sets of coherently controllable open quantum systems with switchable coupling of two models: we show that finite-dimensional systems coupled to a temperature-zero bath allow for approximately generating every quantum state from every initial state. Also studying d-majorization yields an upper bound for non-zero temperature. Then we consider infinite-dimensional open systems following a unital evolution. For arbitrary drift, bounded control Hamiltonians, and noise generated by a compact normal operator, one can approximately reach any state majorized by the initial one.
Quantum information technologies require careful control for generating and preserving a desired target quantum state. The biggest practical obstacle is of course decoherence. Therefore, the reachability analysis, which in our scenario aims to estimate the distance between the controlled state under decoherence and the target state, is of great importance to evaluate the realistic performance of those technologies. This paper presents a lower bound of the fidelity-based distance for a general open Markovian quantum system driven by the decoherence process and several types of control including feedback. The lower bound is straightforward to calculate and can be used as a guide for choosing the target state, as demonstrated in some examples. Moreover, the lower bound is applied to derive a theoretical limit in some quantum metrology problems based on a large-size atomic ensemble under control and decoherence.
Modern applications of quantum control in quantum information science and technology require the precise characterization of quantum states and quantum channels. In particular, high-performance quantum state engineering often demands that quantum states are transferred with optimal efficiency via realizable controlled evolution, the latter often modeled by quantum channels. When an appropriate quantum control model for an interested system is constructed, the exploration of optimal bounds on state transfer for the underlying quantum channel is then an important task. In this work, we analyze the state transfer efficiency problem for different classes of quantum channels, including unitary, mixed unitary, and Markovian. We then apply the theory to nuclear magnetic resonance (NMR) experiments. We show that two most commonly used control techniques in NMR, namely gradient field control and phase cycling, can be described by mixed unitary channels. Then we show that employing mixed unitary channels does not extend the unitarily accessible region of states. Also, we present a strategy of optimal experiment design, which incorporates coherent radio-frequency field control, gradient field control, and phase cycling, aiming at maximizing state transfer efficiency and meanwhile minimizing the number of experiments required. Finally, we perform pseudopure state preparation experiments on two- and three-spin systems, in order to test the bound theory and to demonstrate the usefulness of nonunitary control means.
In quantum systems theory one of the fundamental problems boils down to: Given an initial state, which final states can be reached by the dynamic system in question? Formulated in the framework of bilinear control systems, the evolution shall be governed by an inevitable Hamiltonian drift term, finitely many control Hamiltonians allowing for (at least) piecewise constant control amplitudes, plus a (possibly bang-bang switchable) noise term in Kossakowski-Lindblad form. Now assuming switchable coupling of finite-dimensional systems to a thermal bath of arbitrary temperature, the core problem of reachability boils down to studying points in the standard simplex amenable to two types of controls that can be used interleaved: Permutations within the simplex, and contractions by a dissipative one-parameter semigroup. We illustrate how the solutions of the core problem pertain to the reachable set of the original controlled Markovian quantum system. This allows us to show that for global as well as local switchable coupling to a temperature-zero bath one can generate every quantum state from every initial state up to arbitrary precision. Moreover we present an inclusion for non-zero temperatures as a consequence of our results on d-majorization. Then we consider infinite-dimensional open quantum-dynamical systems following a unital Kossakowski-Lindblad master equation extended by controls. Here the drift Hamiltonian can be arbitrary, the finitely many control Hamiltonians are bounded, and the switchable noise term is generated by a single compact normal operator. Via new majorization results of ours, we show that such bilinear quantum control systems allow to approximately reach any target state majorized by the initial one, as up to now only has been known in finite-dimensional analogues.
Which quantum states can be reached by controlling open Markovian nnn-level quantum systems? Here, we address reachable sets of coherently controllable quantum systems with switchable coupling to a thermal bath of temperature TTT. The core problem reduces to a toy model of studying points in the standard simplex allowing for two types of controls: (i) permutations within the simplex, (ii) contractions by a dissipative semigroup. By illustration, we put the problem into context and show how toy-model solutions pertain to the reachable set of the original controlled Markovian quantum system. Beyond the case T=0T=0T=0 (amplitude damping) we present new results for $0