Procedures for Converting among Lindblad, Kraus and Matrix Representations of Quantum Dynamical Semigroups
Timothy F. Havel
备注：
RevTeX4, 31 pages, no figures; v4 adds new introduction and a numerical example illustrating the application of these results to Quantum Process Tomography
Given an quantum dynamical semigroup expressed as an exponential superoperator acting on a space of N-dimensional density operators, eigenvalue methods are presented by which canonical Kraus and Lindblad operator sum representations can be computed. These methods provide a mathematical basis on which to develop novel algorithms for quantum process tomography, the statistical estimation of superoperators and their generators, from a wide variety of experimental data. Theoretical arguments and numerical simulations are presented which imply that these algorithms will be quite robust in the presence of random errors in the data.
We show that the Truncated Wigner Approximation developed in the flat phase-space is mapped into a Lindblad-type evolution with an indefinite metric in the space of linear operators. As a result, the classically evolved Wigner function corresponds to a non-positive operator $\hat{R}(t)$, which does not describe a physical state. The rate of appearance of negative eigenvalues of $\hat{R}(t)$ can be efficiently estimated. The short-time dynamics of the Kerr and second harmonic generation Hamiltonains are discussed.
The study of dissipation and decoherence in generic open quantum systems recently led to the investigation of spectral and steady-state properties of random Lindbladian dynamics. A natural question is then how realistic and universal those properties are. Here, we address these issues by considering a different description of dissipative quantum systems, namely, the discrete-time Kraus map representation of completely positive quantum dynamics. Through random matrix theory (RMT) techniques and numerical exact diagonalization, we study random Kraus maps, allowing for a varying dissipation stren…
We propose a way to understand the evolution of an open quantum system using a description that dispenses a continuous evolution in time, by discrete operators entangled states, in its most direct and fundamental way. We show that the successive application of these operators in very small time intervals reproduce continuous evolution. It describes and compares the temporal evolution of an open quantum system of three levels, for which the Lindblad equation is solved to obtain the density matrix function of time, a method is developed to find Kraus operators time dependent and also finds const…
Finding the Kraus decomposition from a master equation and vice versa
对于任何在时间上是局部的主方程，无论是Markovian的，非Markovian的，Lindblad形式的或不，一般的程序进行审查，用于构建相应的线性映射从初始状态到状态在时间t，包括其克劳斯型表示。形式上，这相当于求解主方程。对于N维希尔伯特空间，它需要（i）求解一阶N^2 x N^2矩阵的时间演化（以获得完全正映射），以及（ii）对角化相关的N^2 x N^2矩阵（以获得克劳斯型表示）。相反，对于给定的含时线性映射，给出了存在相应主方程的充分必要条件，其中该方程的形式（不一定是唯一的）是明确确定的。它示出了一个“最好的可能”的主方程可能总是被定义，近似的情况下，没有确切的主方程存在的演变。给出了涉及量子位的例子。
For discrete-time systems, governed by Kraus maps, the work of D. Petz has characterized the set of universal contraction metrics. In the present paper, we use this characterization to derive a set of quadratic Lyapunov functions for continuous-time systems, governed by Lindblad differential equations, that have a steady-state with full rank. An extremity of this set is given by the Bures metric, for which the quadratic Lyapunov function is obtained by inverting a Sylvester equation. We illustrate the method by providing a strict Lyapunov function for a Lindblad equation designed to stabilize …
Kraus map closed-form solution for general master equation dynamics
Institute for Theoretical Physics, University of Cologne（科隆大学理论物理研究所）；Forschungszentrum Jülich GmbH, Peter Grünberg Institute, Quantum Control (PGI-8)（于利希研究中心有限公司彼得·格林伯格研究所量子控制部门（PGI-8））
Noisy quantum circuit simulation with the tensor jump method
Weierstrass Institute for Applied Analysis and Stochastics（魏尔斯特拉斯应用分析与随机性研究所）；Technical University of Munich（慕尼黑工业大学）；Munich Quantum Software Company GmbH（慕尼黑量子软件有限公司）；Software Competence Center Hagenberg GmbH (SCCH)（哈根贝格软件能力中心有限公司）
We consider the exact time-evolution of a broad class of fermionic open quantum systems with both strong interactions and strong coupling to wide-band reservoirs. We present a nontrivial fermionic duality relation between the evolution of states (Schrödinger) and of observables (Heisenberg). We show how this highly nonintuitive relation can be understood and exploited in analytical calculations within all canonical approaches to quantum dynamics, covering Kraus measurement operators, the Choi-Jamiołkowski state, time-convolution and convolutionless quantum master equations and generalized Lind…
We develop a microscopic theory for biasing the quantum trajectories of an open quantum system, which renders rare trajectories typical. To this end we consider a discrete-time quantum dynamics, where the open system collides sequentially with qubit probes which are then measured. A theoretical framework is built in terms of thermodynamic functionals in order to characterize its quantum trajectories (each embodied by a sequence of measurement outcomes). We show that the desired biasing is achieved by suitably modifying the Kraus operators describing the discrete open dynamics. From a microscop…
Low-rank propagation for tridiagonalizable open quantum systems: near-linear scaling with system size
Akhiezer Institute for Theoretical Physics, NSC KIPT（阿赫伊泽尔理论物理研究所，NSC KIPT）；Advanced Photonic and Electronic Sciences Division, U.S. Army DEVCOM Army Research Laboratory（美国陆军DEVCOM陆军研究实验室先进光子与电子科学部）；Department of Physics, University of Massachusetts at Boston（马萨诸塞大学波士顿分校物理系）；Education and Research Institute “School of Physics and Technology”, Karazin Kharkiv National University（哈尔科夫国立大学教育与研究所“物理与技术学院”）；Department of Physics and Engineering Physics, Tulane University（杜兰大学物理与工程物理系）
This article provides a review of recent developments in the formulation and execution of optimal control strategies for the dynamics of quantum systems. A brief introduction to the concept of optimal control, the dynamics of of open quantum systems, and quantum information processing is followed by a presentation of recent developments regarding the two main tasks in this context: state-specific and state-independent optimal control. For the former, we present an extension of conventional theory (Pontryagin's principle) to quantum systems which undergo a non-Markovian time-evolution. Owing to…
Stochastic Krylov Dynamics: Revisiting Operator Growth in Open Quantum Systems
Indian Institute of Technology Gandhinagar（印度理工学院甘德纳加尔分校）；University of Cape Town（开普敦大学）；National Institute of Theoretical and Computational Sciences (NITheCS)（国家理论与计算科学研究所）
