D. Salgado, J. L. Sanchez-Gomez
Using the Bloch representation of an N-dimensional quantum system and immediate results from quantum stochastic calculus, we establish a closed formula for the Bloch vector, hence also for the density operator, of a quantum system following a Lindblad evolution with selfadjoint Lindblad operators.
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The relationship between the classic magnetic resonance density matrix relaxation theories of Bloch and Hubbard, and the modern Lindbladian master equation methods are explored. These classic theories are in full agreement with the latest results obtained by the modern methods. A careful scrutiny shows that this also holds true for Redfields later treatment, offered in 1965. The early contributions of Bloch and Hubbard to rotating frame relaxation theory are also highlighted. Taken together, these seminal efforts of Bloch and Hubbard can enjoy a new birth of contemporary relevance in magnetic …
Taming the Bloch-Redfield equation: Recovering an accurate Lindblad equation for general open quantum systems
针对 Bloch-Redfield 方程不能保证完全正性的问题，本文修正已有 Lindblad 变换中的非厄米能移与负衰减率，提出保留久期近似外效应的通用构造方法。
From the Bloch equation to a thermodynamically consistent master equation
本研究针对开放量子系统中无约束Lindblad主方程热力学不一致、缺乏物理解释的问题，提出严格能量守恒下等价的元素布洛赫方程，分解动力学过程以恢复物理直觉，并推导多能级形式、验证其应用。
The perturbative master equation (Bloch-Redfield) is extensively used to study dissipative quantum mechanics - particularly for qubits - despite the 25 year old criticism that it violates positivity (generating negative probabilities). We take an arbitrary system coupled to an environment containing many degrees-of-freedom, and cast its perturbative master equation (derived from a perturbative treatment of Nakajima-Zwanzig or Schoeller-Schon equations) in the form of a Lindblad master equation. We find that the equation's parameters are time-dependent. This time-dependence is rarely accounted …
Understanding the physics of D-Wave annealers: From Schrödinger to Lindblad to Markovian Dynamics
本研究分析了D-Wave退火炉的采样行为，通过标准与快速退火协议结合Bloch方程、Lindblad和Markovian主方程模拟，验证了其结果与Gibbs分布的一致性，揭示了量子力学在其中的作用。
Lindblad equation for the quantum Rabi model in the deep strong coupling regime
本文在深强耦合区推导了量子拉比模型的林德布拉德主方程，结合久期与奇异耦合近似，经基准验证与Bloch-Redfield方程高度一致，并给出极限解析结果。
We develop a Markovian master equation in the Lindblad form that enables the efficient study of a wide range of open quantum many-body systems that would be inaccessible with existing methods. The validity of the master equation is based entirely on properties of the bath and the system-bath coupling, without any requirements on the level structure within the system itself. The master equation is derived using a Markov approximation that is distinct from that used in earlier approaches. We provide a rigorous bound for the error induced by this Markov approximation; the error is controlled by a…
Optimal Control of a Markovian Qubit with Unitary Control
该研究针对受快速幺正控制的马尔可夫量子比特系统，结合约化控制系统与最优控制理论，得到时间最优的冷却/加热控制策略与布洛赫球可稳定态集合，引入解形式简洁的积分系统，方法以解析为主、可直观可视化。
A comparison of different master equations for driven-dissipative dynamics in composite quantum systems: Dispersive readout in structured electromagnetic environments
本文通过微观Bloch-Redfield方法比较了Lindblad和Redfield主方程在驱动-耗散量子比特-谐振子系统中的差异，揭示了无驱动时衰减率的定量不同以及驱动强度下时间相关Redfield耗散器的定性新行为，并研究了结构化谱密度中驱动对测量诱导弛豫的抑制。
We discuss mapping the Bloch-Redfield master-equation to Lindblad form and then unravelling the resulting evolution into a stochastic Schrödinger equation according to the quantum-jump method. We give two approximations under which this mapping is valid. This approach enables us to study solid-state-systems of much larger sizes than is possible with the standard Bloch-Redfield master-equation, while still providing a systematic method for obtaining the jump operators and corresponding rates. We also show how the stochastic unravelling of the Bloch-Redfield equations becomes the kinetic Monte C…
The hidden negative differential thermal conductance
针对量子热输运中Lindblad动力学对耦合原子热流的预测局限，本研究采用Bloch-Redfield主方程发现非平衡系统存在热流抑制即隐藏的负微分热导，证明超越Lindblad的动力学对准确捕捉量子热力学非线性特征至关重要。
In the Bloch sphere picture, one finds the coefficients for expanding a single-qubit density operator in terms of the identity and Pauli matrices. A generalization to $n$ qubits via tensor products represents a density operator by a real vector of length $4^n$, conceptually similar to a statevector. Here, we study this approach for the purpose of quantum circuit simulation, including noise processes. The tensor structure leads to computationally efficient algorithms for applying circuit gates and performing few-qubit quantum operations. In view of variational circuit optimization, we study ``b…
Realistic models of quantum systems must include dissipative interactions with an environment. For weakly-damped systems the Lindblad-form Markovian master equation is invaluable for this task due to its tractability and efficiency. This equation only applies, however, when the frequencies of any subset of the system's transitions are either equal (degenerate), or their differences are much greater than the transitions' linewidths (far-detuned). Outside of these two regimes the only available efficient description has been the Bloch-Redfield (B-R) master equation, the efficacy of which has lon…
Lindblad Superoperators from Wigner's Phase Space Continuity Equation
本文利用 Wigner 1932年的相空间表述，从简单量子系统与环境的弱耦合连续性方程中必然推导出 Lindblad 形式的耦合项。
Markovian dynamics of open quantum systems are described by the L-GKS equation, known also as the Lindblad equation. The equation is expressed by means of left and right matrix multiplications. This formulation hampers numerical implementations. Representing the dynamics by a matrix-vector notation overcomes this problem. We review three approaches to obtain such a representation. The methods are demonstrated for a driven two-level system subject to spontaneous emission.
The article considers a two-level open quantum system, whose evolution is governed by the Gorini--Kossakowski--Lindblad--Sudarshan master equation with Hamiltonian and dissipation superoperator depending, correspondingly, on piecewise constant coherent and incoherent controls with constrained magnitudes. Additional constraints on controls' variations are also considered. The system is analyzed using Bloch parametrization of the system's density matrix. We adapt the section method for obtaining outer parallelepipedal and pointwise estimations of reachable and controllability sets in the Bloch b…
We formulate the general approach based on the Lindblad equation to calculate the full counting statistics of work and heat produced by driven quantum systems weakly coupled with a Markovian thermal bath. The approach can be applied to a wide class of dissipative quantum systems driven by an arbitrary force protocol. We show the validity of general fluctuation relations and consider several generic examples. The possibilities of using calorimetric measurements to test the presence of coherence and entanglement in the open quantum systems are discussed.
A quantum algorithm to simulate Lindblad master equations
本文提出一种基于二阶乘积公式和随机编译的量子算法，用于模拟 Lindblad 主方程，无需辅助量子比特即可实现耗散过程并降低门复杂度，同时扩展至时间依赖情形并给出新的 diamond 范数误差界。
Lindblad Multiproduct Formulas
提出量子误差 mitigation 技术林德布拉德多乘积公式，融入Clifford重标度技术，在65量子比特量子计算机应用获最高5.6倍GPU加速。
Lindblad evolution as gradient flow
本文论证了对于一大类跳跃算符，Lindblad演化可表示为密度算符空间中的梯度流，给出了特征值与特征向量的显式表达式，并证明非厄米情形下演化仍具梯度流特性且稳态由势决定。
We use the Bloch-Redfield-Wangsness theory to calculate the effects of acoustic phonons in coherent control experiments, where quantum-dot excitons are driven by shaped laser pulses. This theory yields a generalized Lindblad equation for the density operator of the dot, with time-dependent damping and decoherence due to phonon transitions between the instantaneous dressed states. It captures similar physics to the form recently applied to Rabi oscillation experiments [A. J. Ramsay et al., Phys. Rev. Lett. 104, 017402 (2010)], but guarantees positivity of the density operator. At sufficiently l…
We previously remarked that when an observable A has a continuous spectrum, then von Neumann's formula for the post-measurement state needs to be extended and the correct formula ineluctably involves the resolution of the detector used in the measurement. We generalize previous results to compute the uncertainties in successive measurements of more general pairs of observables. We also show that this extended von Neumann's formula for the post-measurement state is a completely positive map and, moreover, that there is a completely-positive interpolation between the pre- and post-measurement st…
We study open quantum systems whose evolution is governed by a master equation of Kossakowski-Gorini-Sudarshan-Lindblad type and give a characterization of the convex set of steady states of such systems based on the generalized Bloch representation. It is shown that an isolated steady state of the Bloch equation cannot be a center, i.e., that the existence of a unique steady state implies attractivity and global asymptotic stability. Necessary and sufficient conditions for the existence of a unique steady state are derived and applied to different physical models including two- and four-level…
A Fractional Calculus Framework for Open Quantum Dynamics: From Liouville to Lindblad to Memory Kernels
本文提出了一种分数微积分框架，用于描述具有长记忆效应的开放量子系统动力学，将分数主方程嵌入到更广泛的开放系统形式化体系中，为非马尔可夫性提供了新的理论工具。
In recent years, the Renyi entropy has repeatedly been discussed for characterization of quantum critical states and entanglement. Here, time evolution of the Renyi entropy is studied. A compact general formula is presented for the lower bound on the entropy rate.
So far there is no closed formula for relative entropy of entanglement of arbitrary two-qubit states. In this paper we present a method, which guarantees the derivation of the relative entropy of entanglement for most states that have $z$-directional Bloch vectors. It is shown that the closest separable states for those states also have $z$-directional Bloch vectors though there are few exceptions.
Towards robust variational quantum simulation of Lindblad dynamics via stochastic Magnus expansion
本文提出基于随机Magnus展开与指数积分器的高阶变分量子模拟框架，用于求解Lindblad方程，在大步长稳定性和减少所需量子轨迹数量方面提升鲁棒性，并通过TFIM、FMO和RPM等模型验证其有效性。
A Lindblad-Pauli Framework for Coarse-Grained Chaotic Binary-State Dynamics
本文提出一种基于Lindblad框架的二态模型，用于描述混沌双稳振子的粗粒化动力学，通过密度矩阵方法实现对状态空间的建模与分析。
In this study for particular states of bipartite quantum system in 2n?2m dimensional Hilbert space state, similar to m or n-qubit density matrices represented in Bloch sphere we call them generalized Bloch sphere states(GBSS), we give an efficient optimization procedure so that analytic evaluation of quantum discord can be performed. Using this optimization procedure, we find an exact analytical formula for the optimum positive operator valued measure (POVM) that maximize the measure of the classical correlation for these states. The presented optimization procedure also is used to show that f…
Non-Hermitian Generalization of Bloch Sphere in Spacetime Algebra
该研究利用时空代数建立了非厄米两能级量子系统的布洛赫球几何推广，将态空间扩展为未来光锥，分类了哈密顿量动力学类型并研究了PT对称量子力学的异常点拓扑特征。
Solids can be cooled by driving impurity ions with lasers, allowing them to transfer heat from the lattice phonons to the electromagnetic surroundings. This exemplifies a quantum thermal machine, which uses a quantum system as a working medium to transfer heat between reservoirs. We review the derivation of the Bloch-Redfield equation for a quantum system coupled to a reservoir, and its extension, using counting fields, to calculate heat currents. We use the full form of this equation, which makes only the weak-coupling and Markovian approximations, to calculate the cooling power for a simple …
From Majorana to Bloch: two ways to generalize the Bloch sphere representation
该研究对比纯qudit态的马约拉纳表示与广义布洛赫表示，推导两者间的解析关系，还分析其在对称多量子比特态的扩展，为研究两量子比特态的量子关联提供几何视角。
The absolutely maximally entangled (AME) states play key roles in quantum information processing. We provide an explicit expression of the generalized Bloch representation of AME states for general dimension $d$ of individual subsystems and arbitrary number of partite $n$. Based on this analytic formula, we prove that the trace of the squared support for any given weight is given by the so-called hyper-geometric function and is irrelevant with the choices of the subsystems. The optimal point for the existence of AME states is obtained.
We derive an analog of the Landau-Zener adiabatic tunneling formula for an open, two-level system coupled to a memoryless, dephasing bath. The derivation rests on a geometric view of the spectral subspaces as adiabatic invariants.
So far the Bloch-Siegert shift of the Rabi model has only been calculated numerically or by means of perturbation theory valid in either the weak or strong driving regime only. Recently Yan, Lü, and Zheng [Phys.~Rev.~A {\bf 91}, 053834 (2015)] showed how to reduce the problem to solving a system of three nonlinear equations. Here, we pursue an alternative approach based on a perturbation expansion extrapolation technique. We are thus able to derive an explicit analytical approximation formula for the Bloch-Siegert shift of the Rabi model which is valid for all parameter regimes from weak to st…
We consider an open quantum system which contains unstable states. The time evolution of the system can be described by an effective non-hermitian Hamiltonian H_{eff}, in accord with the Wigner--Weisskopf approximation, and an additional term of the Lindblad form, the socalled dissipator. We show that, after enlarging the original Hilbert space by states which represent the decay products of the unstable states, the non-hermitian part of H_{eff} --the ``particle decay''-- can be incorporated into the dissipator of the enlarged space via a specific Lindblad operator. Thus the new formulation of…
A Randomized Method for Simulating Lindblad Equations and Thermal State Preparation
本文提出了一种随机化方法用于模拟 Lindblad 方程和高效制备热态，通过分解生成器并利用随机采样的跃迁算子，提高了量子模拟的效率。
Quasi-Lindblad pseudomode theory for open quantum systems
本文提出准Lindblad赝模理论，通过放宽完全正性要求并用复指数加权和表示浴关联函数，以紧凑赝模集高效模拟开放量子系统动力学，在玻色子和费米子问题中均优于传统赝模方法。
Hybrid Lindblad dynamics and Bayesian inference from stochastic processes
该研究建立了混合量子-经典动力学的贝叶斯公式化，将量子-经典状态估计转化为经典隐态推断问题，利用粒子滤波等方法近似相关量，平滑可改善隐轨迹重建，框架关联了量子滤波等与贝叶斯时间序列推断机制。
Semigroups describing the time evolution of open quantum systems in finite dimensional spaces have generators of a special form, known as Lindblad generators. The simple generators, characterized by only one operator, are analyzed. The complete set of all the stationary states is presented in detail, including a formula to calculate a stationary state from the generating operator. Also the opposite task can be fulfilled, to construct an evolution leading to a prescribed stationary state.
Markovian reservoir engineering, in which time evolution of a quantum system is governed by a Lindblad master equation, is a powerful technique in studies of quantum phases of matter and quantum information. It can be used to drive a quantum system to a desired (unique) steady state, which can be an exotic phase of matter difficult to stabilize in nature. It can also be used to drive a system to a unitarily-evolving subspace, which can be used to store, protect, and process quantum information. In this paper, we derive a formula for the map corresponding to asymptotic (infinite-time) Lindbladi…
We show that the dynamics of a driven quantum system weakly coupled to a finite reservoir can be approximated by a sequence of Landau-Zener transitions if the level spacing of the reservoir is large enough. This approximation can be formulated as a repeated interaction dynamics and leads to a quantum master equation for the driven system which is of Lindblad form. The approach is validated by comparison with the numerically exact full system dynamics. To emphasize the role of coherence in the master equation, we propose a model-system which shows that in its presence, work can be extracted fro…
We consider an open quantum system described by a Lindblad-type master equation with two times-scales. The fast time-scale is strongly dissipative and drives the system towards a low-dimensional decoherence-free space. To perform the adiabatic elimination of this fast relaxation, we propose a geometric asymptotic expansion based on the small positive parameter describing the time-scale separation. This expansion exploits geometric singular perturbation theory and center-manifold techniques. We conjecture that, at any order, it provides an effective slow Lindblad master equation and a completel…
Qubit-oscillator relationships in the open quantum Rabi model: the role of dissipation
该研究针对耗散开放量子Rabi模型，推导量子比特布洛赫坐标与振荡器可观测量的函数关系，通过多类模拟验证其在强耦合内的有效性，分析了浴耦合对计算的简化作用及超强耦合下非马尔可夫效应带来的计算困难。
Bringing multilevel quantum master equations into Lindblad form for complete positivity tests: Two approaches
本文提出两种由任意 Markov 型多能级量子主方程的 Liouvillian 直接计算 Kossakowski 矩阵的方法，并用其检验 Bloch-Redfield 方程的完全正性，揭示不同三能级系统耗散动力学的相似性。
Linear response and exact hydrodynamic projections in Lindblad equations with decoupled Bogoliubov hierarchies
研究解耦Bogoliubov层次的无自旋费米Lindblad方程，揭示其晚期时间行为为扩散动力学，导致无限热态，并探讨其线性响应函数和流体投影。
Green's functions are fundamental quantities that determine the linear responses of physical systems. The recent developments of non-Hermitian systems, therefore, call for Green's function formulas of non-Hermitian bands. This task is complicated by the high sensitivity of energy spectrums to boundary conditions, which invalidates the straightforward generalization of Hermitian formulas. Here, based on the non-Bloch band theory, we obtain simple Green's function formulas of general one-dimensional non-Hermitian bands. Furthermore, in the large-size limit, these formulas dramatically reduce to …
Algebraic structures of the Lindblad equation
本文揭示有限维开放量子系统Lindblad方程背后的代数结构，通过引入Liouville超算符的算子表示，将动力学表述为与模型无关的Hermitian算子闭代数，并推导出递归关系以高效构造代数，从而降低计算成本。
Bound state formation within the Lindblad approach
本文用重新表述为扩散-平流方程的Lindblad方法研究Pöschl-Teller势下非相对论束缚态的形成时间与热平衡，为理解重离子碰撞中氘核的“地狱中的雪球”现象提供流体力学框架。
A topological formulation of the eigenspace anholonomy, where eigenspaces are interchanged by adiabatic cycles, is introduced. The anholonomy in two-level systems is identified with a disclination of the director (headless vector) of a Bloch vector, which characterizes eigenprojectors. The covering map structure behind the exotic quantum holonomy and the role of the homotopy classification of adiabatic cycles are elucidated. The extensions of this formulation to nonadiabatic cycles and N-level systems are outlined.
Non-Bloch Quantum Geometry of Non-Hermitian Systems
本文提出了一种适用于非厄米特系统的量子几何框架，通过实空间和非布里渊表示中的量子几何量定义，建立了超越传统布洛赫带理论的统一框架。核心成果是实空间积分量子度量与非布里渊区定义的积分量子度量之间的精确等价关系，并引入了非布里渊局域Wannier函数，展示了非布里渊积分量子度量在皮肤效应下的规范不变部分。
We study the Riemannian metric and the Euler characteristic number of the Bloch band in a 1D spin model with multi-site spins exchange interactions. The Euler number of the Bloch band originates from the Gauss-Bonnet theorem on the topological characterization of the closed Bloch states manifold in the first Brillouin zone. We study this approach analytically in a transverse field XY spin chain with three-site spin coupled interactions. We define a class of cyclic quantum distance on the Bloch band and on the ground state, respectively, as a local characterization for quantum phase transitions…
The dynamics of Gaussian states for open quantum systems described by Lindblad equations can be solved analytically for systems with quadratic Hamiltonians and linear Lindbladians, showing the familiar phenomena of dissipation and decoherence. It is well known that the Lindblad dynamics can be expressed as an ensemble average over stochastic pure-state dynamics, which can be interpreted as individual experimental implementations, where the form of the stochastic dynamics depends on the measurement setup. Here we consider quantum-jump and stochastic Schrödinger dynamics for initially Gaussian s…
Bound-state formation and thermalization within the Lindblad approach
本文利用Lindblad动力学及其扩散-对流流体力学重构，研究一维类Pöschl-Teller势中粒子与热浴相互作用下的束缚态形成、热化时间尺度及多束缚态情形，并讨论退相干与熵产生等开放量子系统一般问题。
Boltzmann to Lindblad: Classical and Quantum Approaches to Out-of-Equilibrium Statistical Mechanics
本文提出了一种将经典随机动力学扩展到量子领域的框架，通过分析谐振子动力学，确定了在哈密顿方程中包含摩擦和噪声的条件，以确保完全正性，并揭示了摩擦系数的普遍正性条件。
A semiclassical approximation for an evolving density operator, driven by a "closed" hamiltonian operator and "open" markovian Lindblad operators, is obtained. The theory is based on the chord function, i.e. the Fourier transform of the Wigner function. It reduces to an exact solution of the Lindblad master equation if the hamiltonian operator is a quadratic function and the Lindblad operators are linear functions of positions and momenta. Initially, the semiclassical formulae for the case of hermitian Lindblad operators are reinterpreted in terms of a (real) double phase space, generated by a…
We study the thermalization of an ensemble of $N$ elementary, arbitrarily-complex, quantum systems, mutually noninteracting but coupled as electric or magnetic dipoles to a blackbody radiation. The elementary systems can be all the same or belong to different species, distinguishable or indistinguishable, located at fixed positions or having translational degrees of freedom. Even if the energy spectra of the constituent systems are nondegenerate, as we suppose, the ensemble unavoidably presents degeneracies of the energy levels and/or of the energy gaps. We show that, due to these degeneracies…
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