Quantum Markovian Subsystems: Invariance, Attractivity, and Control

We characterize the dynamical behavior of continuous-time, Markovian quantum systems with respect to a subsystem of interest. Markovian dynamics describes a wide class of open quantum systems of relevance to quantum information processing, subsystem encodings offering a general pathway to faithfully represent quantum information. We provide explicit linear-algebraic characterizations of the notion of invariant and noiseless subsystem for Markovian master equations, under different robustness assumptions for model-parameter and initial-state variations. The stronger concept of an attractive quantum subsystem is introduced, and sufficient existence conditions are identified based on Lyapunov's stability techniques. As a main control application, we address the potential of output-feedback Markovian control strategies for quantum pure state-stabilization and noiseless-subspace generation. In particular, explicit results for the synthesis of stabilizing semigroups and noiseless subspaces in finite-dimensional Markovian systems are obtained.     

用量子广义测量控制消相干

本文探讨了把量子广义测量和无消相干子空间 (DFS) 结合起来抑制消相干的潜力. 证实了: 把量子广义子空间投影测量 (QGSPM) 和量子广义分类投影测量 (QGCPM) 与 DFS 算子条件结合起来, 可以有效增强 Markovian 和非 Markovian 量子开放系统抑制消相干的能力. 强调了量子测量可以作为操控量子态的重要手段. 本方法的优点在于可以构造性地设计相干控制哈密顿量.     

Robust decentralized stabilization of Markovian jump large-scale systems: A neighboring mode dependent control approach

This paper is concerned with the decentralized stabilization problem for a class of uncertain large-scale systems with Markovian jump parameters. The controllers use local subsystem states and neighboring mode information to generate local control inputs. A sufficient condition involving rank constrained linear matrix inequalities is proposed for the design of such controllers. A numerical example is given to illustrate the developed theory.     

Stabilization for Switched LPV Systems with Markovian Jump Parameters and Its Application

This paper is concerned with the stabilization problem for a class of switched linear parameter‐varying (LPV) systems with Markovian jump parameters whose transition rate is completely unknown, or only its estimated value is known. Firstly, a new criterion for testing the stochastic stability of such systems is established. Then, using the multiple parameter‐dependent Lyapunov function method, we design a parameter‐dependent state‐feedback controller for individual switched LPV subsystem to guarantee stochastic stability of the closed‐loop switched LPV systems with Markovian jump parameters under uncertain transition rates. Finally, as an application of the proposed design method, the stabilization problem of a turbofan‐engine which cannot be handled by the existing methods is investigated.     

On hybrid control of a class of stochastic non-linear Markovian switching systems

In this paper, we address the problem of hybrid control for a class of stochastic non-linear Markovian switching systems. First, a hybrid controller is introduced for the systems. Then under some appropriate assumptions, the stabilization condition for the systems under pure impulsive control is given. Further under impulsive control, the output feedback stabilization problem of the systems is discussed and linear output feedback controllers are designed. Finally a numerical example is provided to illustrate the effectiveness of the proposed methods.     

Negativity fonts, multiqubit invariants and four qubit maximally entangled states

Recently, we introduced negativity fonts as the basic units of multipartite entanglement in pure states. We show that the relation between global negativity of partial transpose of N?qubit state and linear entropy of reduced single qubit state yields an expression for global negativity in terms of determinants of negativity fonts. Transformation equations for determinants of negativity fonts under local unitaries (LU??s) are useful to construct LU invariants such as degree four and degree six invariants for four qubit states. The difference of squared negativity and N?tangle is an N qubit invariant which contains information on entanglement of the state caused by quantum coherences that are not annihilated by removing a single qubit. Four qubit invariants that detect the entanglement of specific parts in a four qubit state are expressed in terms of three qubit subsystem invariants. Numerical values of invariants bring out distinct features of several four qubit states which have been proposed to be the maximally entangled four qubit states.     

Global stabilization of partially linear composite systems using homogeneous method

A linear feedback control scheme to globally stabilize a class of partially linear composite systems is proposed from the point view of homogeneity. Assume that the global stability of the zero dynamics of the nonlinear subsystem can be tested by using a homogeneous Lyapunov function. It is shown that the stabilization of the linear controllable subsystem from its own states equals to the stabilization of the whole systems if the nonlinearities satisfy a homogeneous inequality condition. Then we assume that the states are not measurable and also extend the method developed for state‐feedback control to the output‐feedback case. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Discrete-time controllability for feedback quantum dynamics

Controllability properties for discrete-time, Markovian quantum dynamics are investigated. We find that, while in general the controlled system is not finite-time controllable, feedback control allows for arbitrary asymptotic state-to-state transitions. Under further assumptions on the form of the measurement, we show that finite-time controllability can be achieved in a time that is twice the dimension of the system, and we provide an iterative procedure to design the unitary control actions.     

A phase-space formulation and Gaussian approximation of the filtering equations for nonlinear quantum stochastic systems

This paper is concerned with a filtering problem for a class of nonlinear quantum stochastic systems with multichannel nondemolition measurements. The system-observation dynamics are governed by a Markovian Hudson-Parthasarathy quantum stochastic differential equation driven by quantum Wiener processes of bosonic fields in vacuum state. The Hamiltonian and system-field coupling operators, as functions of the system variables, are assumed to be represented in a Weyl quantization form. Using the Wigner-Moyal phase-space framework, we obtain a stochastic integro-differential equation for the posterior quasi-characteristic function (QCF) of the system conditioned on the measurements. This equation is a spatial Fourier domain representation of the Belavkin-Kushner-Stratonovich stochastic master equation driven by the innovation process associated with the measurements. We discuss a specific form of the posterior QCF dynamics in the case of linear system-field coupling and outline a Gaussian approximation of the posterior quantum state.     

Invariance control for a class of cascade nonlinear systems

We consider the control of partially linear cascade systems using switching control of the states of the linear subsystem. We give sufficient conditions under which feedback of the linear states with switching gains guarantees both exponential stability of the linear subsystem and positive invariance of a prespecified region in state space. We refer to a control scheme incorporating these two objectives as invariance control. Semiglobal asymptotic stabilization follows under some additional conditions. The key idea of our design is to keep a given state space region positively invariant by switching on the boundary of the region. Thus, the transient response of the system can be kept within prescribed bounds which is important in many practical applications. Our approach can also be viewed as an alternative to high gain designs. The results in this paper can be extended to nonlinear cascades and even to noncascaded systems     

借助Lyapunov方法的量子系统平衡态的布居控制

This paper studies the population control problem associated with the equilibrium states of mixed-state quantum systems by using a Lyapunov function with degrees of freedom. The control laws are designed by ensuring the monotonicity of the Lyapunov function; main results on the largest invariant set in the sense of LaSalle are given; and the strict expression of any state in the largest invariant set is normally deduced in the framework of Bloch vectors. By analyzing the obtained largest invariant set and the Lyapunov function itself, this paper also discusses the determination problem of the degrees of freedom. Numerical simulation experiments on a three-level system show the validity of research results.     

Improved constructions of mixed state quantum automata

Quantum finite automata with mixed states are proved to be super-exponentially more concise rather than quantum finite automata with pure states. It was proved earlier by A. Ambainis and R. Freivalds that quantum finite automata with pure states can have an exponentially smaller number of states than deterministic finite automata recognizing the same language. There was an unpublished “folk theorem” proving that quantum finite automata with mixed states are no more super-exponentially more concise than deterministic finite automata. It was not known whether the super-exponential advantage of quantum automata is really achievable.     

Semi‐Global Output Feedback Stabilization for a Class of Uncertain Nonlinear Systems

This paper addresses the problem of semi‐global stabilization by output feedback for a class of nonlinear systems whose output gains are unknown. For each subsystem, we first design a state compensator and use the compensator states to construct a control law to stabilize the nominal linear system without the perturbing nonlinearities. Then, combining the output feedback domination approach with block‐backstepping scheme, a series of homogeneous output feedback controllers are constructed recursively for each subsystem and the closed‐loop system is rendered semi‐globally asymptotically stable.     

广义马氏跳变系统在一般转移速率下的控制器设计

针对转移概率不能精确获得并且部分未知的广义马氏跳变系统, 分别讨论在模态依赖控制器和模态独立控制器条件下的系统镇定问题. 与现有结果相比, 所提研究方法具有较小的保守性, 能够有效地解决实际问题. 首先, 运用自由权矩阵方法, 得到了广义马氏跳变系统在转移速率满足上述一般条件时系统随机容许的充分条件. 在此基础上, 以线性矩阵不等式(Linear matrix inequality, LMI)的形式分别给出了模态依赖和模态独立控制器的求解条件. 最后, 通过数值算例验证设计方法的有效性和优越性.     

Stability analysis and saturation control for nonlinear positive Markovian jump systems with randomly occurring actuator faults

This article investigates the stability analysis and control design of a class of nonlinear positive Markovian jump systems with randomly occurring actuator faults and saturation. It is assumed that the actuator faults of each subsystem are varying and governed by a Markovian process. The nonlinear term is located in a sector. First, sufficient conditions for stochastic stability of the underlying systems are established using a stochastic copositive Lyapunov function. Then, a family of reliable L₁‐gain controller is proposed for nonlinear positive Markovian jump systems with actuator faults and saturation in terms of a matrix decomposition technique. Under the designed controllers, the closed‐loop systems are positive and stochastically stable with an L₁‐gain performance. An optimization method is presented to estimate the maximum domain of attraction. Furthermore, the obtained results are developed for general Markovian jump systems. Finally, numerical examples are given to illustrate the effectiveness of the proposed techniques.     

Laplacian matrices of weighted digraphs represented as quantum states

Representing graphs as quantum states is becoming an increasingly important approach to study entanglement of mixed states, alternate to the standard linear algebraic density matrix-based approach of study. In this paper, we propose a general weighted directed graph framework for investigating properties of a large class of quantum states which are defined by three types of Laplacian matrices associated with such graphs. We generalize the standard framework of defining density matrices from simple connected graphs to density matrices using both combinatorial and signless Laplacian matrices associated with weighted directed graphs with complex edge weights and with/without self-loops. We also introduce a new notion of Laplacian matrix, which we call signed Laplacian matrix associated with such graphs. We produce necessary and/or sufficient conditions for such graphs to correspond to pure and mixed quantum states. Using these criteria, we finally determine the graphs whose corresponding density matrices represent entangled pure states which are well known and important for quantum computation applications. We observe that all these entangled pure states share a common combinatorial structure.     

Further Results on the Cross Norm Criterion for Separability

In the present paper we develop and investigate a novel approach that aims to characterize quantum entanglement by using cross norms. In the first part of the paper we further develop the mathematical theory by determining the value of the greatest cross norm for Werner states, for isotropic states and for Bell diagonal states. In the second part we show that our techniques induce a novel powerful analytical and computable separability criterion for bipartite systems. This new criterion complements the well-known Peres positive partial transpose criterion in several aspects. It is a necessary but in general not a sufficient criterion for separability. We demonstrate the power of the new criterion by evaluating the criterion for a number of important examples. We also demonstrate that the new criterion is able to detect bound entangled states.     

Hybrid feedback laws for a class of cascade nonlinear controlsystems

A stabilization problem for a class of nonlinear control systems is considered. Systems in this class can be viewed as a cascade connection of a linear time-invariant subsystem, a nonlinear time-periodic static subsystem, and an integrator. Hybrid logic-based feedback controllers are constructed to globally stabilize these systems to the origin. The controllers operate by switching between various time-periodic control functions at discrete-time instants. As specific applications, we consider stabilization of nonholonomic control systems in power form to the origin and stabilization of trajectories for a class of nonlinear control systems. Numerical examples of global stabilization and tracking are reported     

Beyond complete positivity

We provide a general and consistent formulation for linear subsystem quantum dynamical maps, developed from a minimal set of postulates, primary among which is a relaxation of the usual, restrictive assumption of uncorrelated initial system-bath states. We describe the space of possibilities admitted by this formulation, namely that, far from being limited to only completely positive (CP) maps, essentially any \({\mathbb {C}}\)-linear, Hermiticity-preserving, trace-preserving map can arise as a legitimate subsystem dynamical map from a joint unitary evolution of a system coupled to a bath. The price paid for this added generality is a trade-off between the set of admissible initial states and the allowed set of joint system-bath unitary evolutions. As an application, we present a simple example of a non-CP map constructed as a subsystem dynamical map that violates some fundamental inequalities in quantum information theory, such as the quantum data processing inequality.