Informational correlation between two parties of a quantum system: spin-1/2 chains

We introduce the informational correlation \(E^{AB}\) between two interacting quantum subsystems \(A\) and \(B\) of a quantum system as the number of arbitrary parameters \(\varphi _i\) of a unitary transformation \(U^A\) (locally performed on the subsystem \(A\) ) which may be detected in the subsystem \(B\) by the local measurements. This quantity indicates whether the state of the subsystem \(B\) may be effected by means of the unitary transformation applied to the subsystem \(A\) . Emphasize that \(E^{AB}\ne E^{BA}\) in general. The informational correlations in systems with tensor product initial states are studied in more details. In particular, it is shown that the informational correlation may be changed by the local unitary transformations of the subsystem \(B\) . However, there is some non-reducible part of \(E^{AB}(t)\) which may not be decreased by any unitary transformation of the subsystem \(B\) at a fixed time instant \(t\) . Two examples of the informational correlations between two parties of the four-node spin-1/2 chain with mixed initial states are studied. The long chains with a single initially excited spin (the pure initial state) are considered as well.     

Pairwise quantum correlations of a three-qubit XY chain with phase decoherence

Quantum correlations, including entanglement and discord with its geometric measure in a three-qubit Heisenberg XY chain, with phase decoherence, are investigated when a nonuniform magnetic field is applied. When the qubits are initially in an unentangled state, the nearest neighbor pairwise correlations are destroyed by phase decoherence, but stationary correlations appear for next-to-neighbor qubits. With an inhomogeneous magnetic field, the stationary correlations appear for nearest neighbor qubits and they disappear for next-to-nearest neighbor qubits. But when the qubits are initially in an entangled state, an inhomogeneous magnetic field can enhance the stationary correlations of next-to-neighbor qubits, but it cannot do so for nearest neighbor qubits. The decoherence effect on stationary correlations is much stronger for next-to-nearest neighbor qubits than it is for nearest neighbor qubits. Finally, a uniform magnetic field can affect the correlations when the qubits are initially in an entangled state, but it cannot affect them when the qubits are initially in an unentangled state.     

Evolution of quantum correlations in the open quantum systems consisting of two coupled oscillators

The open quantum systems consisting of coupled and uncoupled asymmetric oscillators are considered with an initial quantum-dot trapped-ion coherent state. The quantum correlations between spatial modes of this trapped ion are examined to find their dependence on the temperature, asymmetric parameter, dissipation coefficient and the magnetic field. It is observed that the discord of the initial state is an increasing function of the asymmetric parameter and the magnetic field. Moreover, in the case of two uncoupled modes, entanglement and discord are decreasing functions of temperature and the dissipation coefficient. However, as the temperature and dissipation coefficient increase, the discord fades out faster. In the case of two coupled modes, as the temperature and dissipation coefficient increase, the sudden death of the entanglement and fade out of the discord happen sooner; moreover, as the magnetic field increases, the entanglement sudden death and the discord fade out time occur sooner. Also, with the increase in the asymmetric parameter, the entanglement sudden death is postponed. In addition, in the asymmetric system, appreciable discord can be created in the temperature range 0–10 K, while appreciable entanglement can be created in the temperature range 0–5 mK. Finally, it is observed that non-monotonic evolution of quantum correlations is due to coupling of modes.     

Thermal entanglement of the mixed spin-1/2 and spin-5/2 Heisenberg model under an external magnetic field

We use the concept of negativity to study the entanglement of spin-1/2 and spin-5/2 antiferromagnetic Heisenberg model with an inhomogeneous magnetic field. Analytical conclusions of the model are acquired. It is found that the critical temperature \(T_\mathrm{c}\) goes up, as the increase of anisotropy parameter \(k\) . The temperature \(T_\mathrm{c}\) becomes bigger than the results of spin-1/2 and spin-3/2 Heisenberg XXZ chain for the same value of \(k\) . And we can gain more entanglement at higher temperature by coordinating the value of inhomogeneity \(b\) .     

Tripartite and bipartite quantum correlations in the XXZ spin chain with three-site interaction

Tripartite and bipartite quantum correlations in the three-qubit XXZ Heisenberg spin chain with two types of three-site interactions and an external magnetic field are investigated. We show that the increase in XZY ? YZX interaction can enhance the robustness of both tripartite and bipartite correlations, whereas the increase in XZX \(+\) YZY interaction could improve the robustness of tripartite quantum correlations, but diminish the robustness of bipartite quantum correlations. Tripartite measurement-induced disturbance is the most robust against temperature, and bipartite entanglement is the most fragile. Tripartite entanglement is even more robust than bipartite quantum discord when XZX \(+\) YZY or XZY ? YZX interaction is relatively large. The cooperative effect of XZX \(+\) YZY and XZY ? YZX interaction could induce bipartite entanglement even at high temperature. The cooperative effect of XZX \(+\) YZY and XZY ? YZX interaction is the most optimal to improve the robustness of all quantum correlations when the magnetic field is negative. When the magnetic field is positive, the effective of XZY ? YZX interaction alone is more ideal to preserve different quantum correlations.     

Problem of quantifying quantum correlations with non-commutative discord

In this work we analyze a non-commutativity measure of quantum correlations recently proposed by Guo (Sci Rep 6:25241, 2016). By resorting to a systematic survey of a two-qubit system, we detected an undesirable behavior of such a measure related to its representation-dependence. In the case of pure states, this dependence manifests as a non-satisfactory entanglement measure whenever a representation other than the Schmidt’s is used. In order to avoid this basis-dependence feature, we argue that a minimization procedure over the set of all possible representations of the quantum state is required. In the case of pure states, this minimization can be analytically performed and the optimal basis turns out to be that of Schmidt’s. In addition, the resulting measure inherits the main properties of Guo’s measure and, unlike the latter, it reduces to a legitimate entanglement measure in the case of pure states. Some examples involving general mixed states are also analyzed considering such an optimization. The results show that, in most cases of interest, the use of Guo’s measure can result in an overestimation of quantum correlations. However, since Guo’s measure has the advantage of being easily computable, it might be used as a qualitative estimator of the presence of quantum correlations.     

Generation and protection of steady-state quantum correlations due to quantum channels with memory

We have proposed a scheme of the generation and preservation of two-qubit steady-state quantum correlations through quantum channels where successive uses of the channels are correlated. Different types of noisy channels with memory, such as amplitude damping, phase damping, and depolarizing channels, have been taken into account. Some analytical or numerical results are presented. The effect of channels with memory on dynamics of quantum correlations has been discussed in detail. The results show that steady-state entanglement between two initial qubits whose initial states are prepared in a specific family states without entanglement subject to amplitude damping channel with memory can be generated. The entanglement creation is related to the memory coefficient of channel \(\mu \). The stronger the memory coefficient of channel \( \mu \) is, the more the entanglement creation is, and the earlier the separable state becomes the entangled state. Besides, we compare the dynamics of entanglement with that of quantum discord when a two-qubit system is initially prepared in an entangled state. We show that entanglement dynamics suddenly disappears, while quantum discord dynamics displays only in the asymptotic limit. Furthermore, two-qubit quantum correlations can be preserved at a long time in the limit of \(\mu \rightarrow 1\).     

Post-Markovian dynamics of quantum correlations: entanglement versus discord

Dynamics of an open two-qubit system is investigated in the post-Markovian regime, where the environments have a short-term memory. Each qubit is coupled to separate environment which is held in its own temperature. The inter-qubit interaction is modeled by XY–Heisenberg model in the presence of spin–orbit interaction and inhomogeneous magnetic field. The dynamical behavior of entanglement and discord has been considered. The results show that quantum discord is more robust than quantum entanglement, during the evolution. Also the asymmetric feature of quantum discord can be monitored by introducing the asymmetries due to inhomogeneity of magnetic field and temperature difference between the reservoirs. By employing proper parameters of the model, it is possible to maintain nonvanishing quantum correlation at high degree of temperature. The results can provide a useful recipe for studying dynamical behavior of two-qubit systems such as trapped spin electrons in coupled quantum dots.     

利用相位的自旋1/2量子系统的相干控制

针对两个自旋1/2粒子组成的封闭量子系统, 建立了具有Ising相互作用的量子系统模型. 在此基础上通过具有特定幅值及相对相位的半反直觉脉冲, 制备了相应的量子相干态. 并通过系统数值仿真实验, 归纳出系统终态与相对相位之间的近似关系式, 分析了控制脉冲的幅值和时间延迟对制备过程的影响. 利用部分绝热通道技术实现了相位相干控制.     

Quotients of Markov chains and asymptotic properties of the stationary distribution of the Markov chain associated to an evolutionary algorithm

In this work, a method is presented for analysis of Markov chains modeling evolutionary algorithms through use of a suitable quotient construction. Such a notion of quotient of a Markov chain is frequently referred to as “coarse graining” in the evolutionary computation literature. We shall discuss the construction of a quotient of an irreducible Markov chain with respect to an arbitrary equivalence relation on the state space. The stationary distribution of the quotient chain is “coherent” with the stationary distribution of the original chain. Although the transition probabilities of the quotient chain depend on the stationary distribution of the original chain, we can still exploit the quotient construction to deduce some relevant properties of the stationary distribution of the original chain. As one application, we shall establish inequalities that describe how fast the stationary distribution of Markov chains modeling evolutionary algorithms concentrates on the uniform populations as the mutation rate converges to 0. Further applications are discussed. One of the results related to the quotient construction method is a significant improvement of the corresponding result of the authors’ previous conference paper [Mitavskiy et al. (2006) In: Simulated Evolution and Learning, Proceedings of SEAL 2006, Lecture Notes in Computer Science v. 4247, Springer Verlag, pp 726–733]. This papers implications are all strengthened accordingly.

Fuzzy Mixed H2/H∞ Optimization-Based Decentralized Model Reference Control and Application to Piezo-Driven XY Table Systems

In this paper, a decentralized model reference control via fuzzy mixed H²/H^infin optimization design was developed. Each subsystem contained L linear pulse transfer function systems (LPTFSs). The reference model for every LPTFS was first designed to shape the response of the ith closed-loop subsystem. Then the H² -norm of the output error (i.e., the difference between the output of the reference model and the system) and weighted control input of the jth LPTFS was minimized to obtain a control such that smaller energy consumption with bounded tracking error of the jth LPTFS was achieved. However, an output disturbance of the jth LPTFS caused by the interactions among the LPTFSs, the interconnections among the subsystems, modeling errors, and external loads deteriorated system performance or even resulted in instability. In this situation, the H ^infin-norm of weighted sensitivity between output disturbance and output error of the jth LPTFS was minimized to attenuate its effect. A nonlinear control based on output error for every LPTFS was also established to improve robust performance. The stability of the overall system was then verified by Lyapunov stability criterion. The application to piezo-driven XY table system (PD-XY-TS) was carried out to confirm the usefulness of the proposed control     

Estimation variance and H∞-errorminimization of stationary process with perfect measurements

The singular estimation problem of linear continuous-time stationary processes is solved both in the minimum error variance and the minimum H_∞-norm sense. Simple explicit expressions for the resulting estimators and their corresponding minimum criteria are derived in terms of the system structure parameters. These expressions enable an easy comparison between the two minimization procedures. The theory is demonstrated by a simple example of fourth order which is solved by both minimization methods     

1/2分数阶线性随机动力系统的非平稳响应解析解

提出随机激励作用下1/2分数阶线性系统非平稳响应解析解的一种新方法.首先,利用特征向量展开得到1/2分数阶阻尼系统的脉冲响应函数解析表达;之后,基于Laplace变换计算得到响应功率谱密度的解析表达式和系统均方响应.通过白噪声、调制白噪声和调制修正金井清谱三种不同随机激励类型的数值算例,利用与蒙特卡洛模拟所得结果对比证明该方法的准确性和适用性.     

Application of fuzzy sets to manufacturing/distribution planning decisions in supply chains

In the real-world manufacturing/distribution planning decision (MDPD) integration problems in supply chains, the environmental coefficients and parameters are normally imprecise due to incomplete and/or unavailable information. This work presents a fuzzy linear programming approach based on the possibility theory. It applies this approach to solve multi-product and multi-time period MDPD problems with imprecise goals and forecast demand by considering the time value of money of related operating cost categories. The proposed approach attempts to minimize the total manufacturing and distribution costs by considering the levels of inventory, subcontracting and backordering, the available machine capacity and labor levels at each source, forecast demand and available warehouse space at each destination. This study utilizes an industrial case study to demonstrate the feasibility of applying the proposed approach to practical MDPD problems. The primary contribution of this paper is a fuzzy mathematical programming methodology for solving the MDPD integration problems in uncertain environments.     

Shape design optimization of stationary fluid-structure interaction problems with large displacements and turbulence

This paper presents general and efficient methods for analysis and gradient based shape optimization of systems characterized as strongly coupled stationary fluid-structure interaction (FSI) problems. The incompressible fluid flow can be laminar or turbulent and is described using the Reynolds-averaged Navier-Stokes equations (RANS) together with the algebraic Baldwin–Lomax turbulence model. The structure may exhibit large displacements due to the interaction with the fluid domain, resulting in geometrically nonlinear structural behaviour and nonlinear interface coupling conditions. The problem is discretized using Galerkin and Streamline-Upwind/Petrov–Galerkin finite element methods, and the resulting nonlinear equations are solved using Newtons method. Due to the large displacements of the structure, an efficient update algorithm for the fluid mesh must be applied, leading to the use of an approximate Jacobian matrix in the solution routine. Expressions for Design Sensitivity Analysis (DSA) are derived using the direct differentiation approach, and the use of an inexact Jacobian matrix in the analysis leads to an iterative but very efficient scheme for DSA. The potential of gradient based shape optimization of fluid flow and FSI problems is illustrated by several examples.     

给定环境下稳定开放量子系统的哈密顿量方法

针对耗散已知情况下Lindblad主方程描述的开放量子系统,本文通过哈密顿量的设计实现了系统对于目标平衡态的稳定化.借助相干矢量体系,将矩阵形式下的原始系统模型转换为了一个矢量形式的线性系统,并证明了变换前后系统稳定属性的等价性.通过保证矢量化线性系统模型的稳定性,并使系统的唯一平衡态等于期望的目标态,得到了系统哈密顿量的设计框架.特别地,本文讨论了这两类条件下系统哈密顿量各元素的范围,并指出根据它们的交集即可构成所设计的系统哈密顿量.最后,在一个两能级系统上进行了数值仿真实验,验证了本文哈密顿量稳定化方案的有效性.     

The solution of stationary ODE problems in quantum mechanics by Magnus methods with stepsize control

In solid state physics the solution of the Dirac and Schrödinger equation by operator splitting methods leads to differential equations with oscillating solutions for the radial direction. For standard time integrators like Runge-Kutta or multistep methods the stepsize is restricted approximately by the length of the period. In contrast the recently developed Magnus methods allow stepsizes that are substantially larger than one period. They are based on a Lie group approach and incorporate exponential functions and matrix commutators. A stepsize control is implemented and tested. As numerical examples eigenvalue problems for the radial Schrödinger equation and the radial Dirac equation are solved. Further, phase shifts for scattering solutions for hydrogen atoms and copper are computed.