E. A. Ivanchenko
he analytic formulas for structure constants of $su(2S+1)$ algebra in terms of $3jm$ and $6j$ symbols of $su(2)$ have been derived for the decomplexification of the Liouville-von Neumann equation.
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We show that Liouville-von Neumann approach to quantum mechanical systems, which demands the existence of invariant operators, reproduces the time-dependent variational Gaussian approximation. We find the effective action of the time-dependent systems and show that many aspects of the dynamics are independent of the details of time evolution, e.g., the squeezing of the wave-function is determined by the effective potential of the final stage of time-evolution.
On the feasibility and usefulness of applying the `Schrödinger c.q. Liouville-von Neumann equation' to quantum measurement
该研究将量子测量建模为客体与仪器微观部分间由Schrödinger方程描述的相互作用，借Stern-Gerlach实验提出区分初态制备与测量相互作用的Heisenberg不等式替代形式。
A new form of a binary Darboux transformation is used to generate analytical solutions of a nonlinear Liouville-von Neumann equation. General theory is illustrated by explicit examples.
The density matrix is a widely used tool in quantum mechanics. In order to determine its evolution with respect to time, the Liouville-von Neumann equation must be solved. However, analytic solutions of this differential equation exist only for simple cases. Additionally, if the equation is coupled to Maxwell's equations to model light-matter interaction, the resulting equation set -- the Maxwell-Bloch or Maxwell-Liouville-von Neumann (MLN) equations -- becomes nonlinear. In these advanced cases, numerical methods are required. Since the density matrix has certain mathematical properties, the …
The Stochastic Liouville-von Neumann equation provides an exact numerical simulation strategy for quantum systems interacting with Gaussian reservoirs [J.T. Stockburger & H. Grabert, PRL 88, 170407 (2002)]. Its scaling with the extension of the time interval covered has recently improved dramatically through time-domain projection techniques [J.T. Stockburger, EPL 115, 40010 (2016)]. Here we present a sampling strategy which results in a significantly improved scaling with the strength of the dissipative interaction, based on reducing the non-unitary terms in sample propagation through convex …
On the Liouville-von Neumann equation for unbounded Hamiltonians
本文针对无界哈密顿量情形，严格且自洽地介绍了量子刘维尔形式体系，发展了 Liouvillian 及其平方定义域的简化刻画，并给出其本质自伴核的显式证明。
We formulate the thermofield dynamics for time-dependent systems by combining the Liouville-von Neumann equation, its invariant operators, and the basic notions of thermofield dynamics. The new formulation is applied to time-dependent bosons and fermions by using the time-dependent annihilation and creation operators that satisfy the Liouville-von Neumann equation. It is shown that the thermal state is the time- and temperature-dependent vacuum state and a general formula is derived to calculate the thermal expectation value of operators.
The evolution of Liouville von Neumann master equations in the Pechukas-Yukawa framework
本文提出基于 Pechukas-Yukawa 映射的本征值动力学形式体系，精确描述密度矩阵非平衡演化，并阐明非平衡量子相变与退相干的关系。
Canonical coordinates for both the Schroedinger and the nonlinear Schroedinger equations are introduced, making more transparent their Hamiltonian structures. It is shown that the Schroedinger equation, considered as a classical field theory, shares with the nonlinear Schroedinger, and more generally with Liouville completely integrable field theories, the existence of a "recursion operator" which allows for the construction of infinitely many conserved functionals pairwise commuting with respect to the corresponding Poisson bracket. The approach may provide a good starting point to get a clea…
An extension of the Liouville-von Neumann dynamics to a Nambu-type dynamics is proposed. The resulting theory is the first version of nonlinear QM which is free from internal inconsistencies.
Classical mechanics (CM), like quantum mechanics (QM), can have an operatorial formulation. This was pioneered by Koopman and von Neumann (KvN) in the 30's. They basically formalized, via the introduction of a classical Hilbert space, earlier work of Liouville who had shown that the classical time evolution can take place via an operator, nowadays known as the Liouville operator. In this paper we study how to perform the coupling of a point particle to a gauge field in the KvN version of CM. So we basically implement at the classical operatorial level the analog of the minimal coupling of QM. …
In a recent article, Chiang and Hsu [The Journal of Chemical Physics 153, 044103 (2020)] examine one and two site electronic junctions identically connected to finite reservoirs. For these two examples, they derive analytical solutions, as well as provide asymptotic analyses, for the steady-state current from the driven Liouville-von Neumann (DLvN) equation - an open system approach to transport in non-interacting systems where relaxation maintains a bias. The two site junction they examine has destructive interference, which they show leads to slow convergence of the DLvN to the Landauer limi…
Quantum computers can be used to simulate nonlinear non-Hamiltonian classical dynamics on phase space by using the generalized Koopman-von Neumann formulation of classical mechanics. The Koopman-von Neumann formulation implies that the conservation of the probability distribution function on phase space, as expressed by the Liouville equation, can be recast as an equivalent Schrödinger equation on Hilbert space with a Hermitian Hamiltonian operator and a unitary propagator. This Schrödinger equation is linear in the momenta because it derives from a constrained Hamiltonian system with twice th…
Koopman and von Neumann (KvN) extended the Liouville equation by introducing a phase space function $S^{(K)}(q,p,t)$ whose physical meaning is unknown. We show that a different $S(q,p,t)$, with well-defined physical meaning, may be introduced without destroying the attractive "quantum-like" mathematical features of the KvN theory. This new $S(q,p,t)$ is the classical action expressed in phase space coordinates. It defines a mapping between observables and operators which preserves the Lie bracket structure. The new evolution equation reduces to Schrödinger's equation if functions on phase spac…
The von Neumann evolution equation for density matrix and the Moyal equation for the Wigner function are mapped onto evolution equation for optical tomogram of quantum state. The connection with known evolution equation for symplectic tomogram of the quantum state is clarified. The stationary states corresponding to quantum energy levels are associated with the probability representation of the von Neumann and Moyal equations written for the optical tomograms. Classical Liouville equation for optical tomogram is obtained. Example of parametric oscillator is considered in detail.
We review some recent developments in the theory of nonlinear von Neumann equations. We distinguish between the von Neumann equation (which can be nonlinear) and the Liouville equation (which should be linear). Explicit examples illustrate the technique of binary Darboux integration of nonlinear density matrix equations and special attention is payed to the problem of how to find physically nontrivial `self-scattering' solutions.
Implementing the Koopman-von Neumann approach on continuous-variable photonic quantum computers
本文探讨在连续变量光子量子计算架构上实现KvN方法，用于模拟复杂非线性动力学。
We present a partition-free approach to the evolution of density matrices for open quantum systems coupled to a harmonic environment. The influence functional formalism combined with a two-time Hubbard-Stratonovich transformation allows us to derive a set of exact differential equations for the reduced density matrix of an open system, termed the Extended Stochastic Liouville-von Neumann equation. Our approach generalises previous work based on Caldeira-Leggett models and a partitioned initial density matrix. This provides a simple, yet exact, closed-form description for the evolution of open …
Solutions of Koopman-von Neumann equations, their superpositions, and uncertainties
本文研究了KvN方程的解及其叠加、正交性和不确定性，探讨了经典系统中不确定性关系的产生及本征态的正交性。
A generalization of the Gibbs-von Neumann relative entropy is proposed based on the quantum BBGKY [Bogolyubov-Born-Green-Kirkwood-Yvon] hierarchy as the nonequilibrium entropy for an N-body system. By using a generalization of the Liouville-von Neumann equation describing the evolution of a density super- operator, it is demonstrated that the entropy production for an isolated system is non-negative, which provides an arrow of time. Moreover, following the procedure of non-equilibrium thermodynamics a master matrix is introduced for which a mi- croscopic expression is obtained. Then, the quant…
Combination of the Liouville equation with the q-averaged energy $U_q = <H>_q$ leads to a microscopic framework for nonextensive q-thermodynamics. The resulting von Neumann equation is nonlinear: $i\dotρ=[H,ρ^q]$. In spite of its nonlinearity the dynamics is consistent with linear quantum mechanics of pure states. The free energy $F_q=U_q-TS_q$ is a stability function for the dynamics. This implies that q-equilibrium states are dynamically stable. The (microscopic) evolution of $ρ$ is reversible for any q, but for $q\neq 1$ the corresponding macroscopic dynamics is irreversible.
Starting from the Liouville-von Neumann equation, under a weak coupling limit we derive the Lindblad master equation for the one-dimensional quantum Ising model in a Markov approximation and a rotating wave approximation. We also prove that the steady solution of the Lindblad equation is the canonical distribution independent of the dissipation rate.
We derive a master equation from the exact stochastic Liouville von-Neumann (SLN) equation \cite{stockburger2002,wallsbook}. The latter depends on two correlated noises and describes exactly the dynamics of an oscillator (which can be either harmonic or present an anharmonicity) coupled to an environment at thermal equilibrium. The newly derived master equation is obtained by performing analytically the average over different noise trajectories. It is found to have a complex hierarchical structure that might be helpful to explain the convergence problems occurring when performing numerically t…
On the numerical limitations of dual Koopman von Neumann embeddings for solving conservative nonlinear ordinary differential equations on quantum computers
本文提出基于Koopman-von Neumann嵌入的量子算法求解非线性常微分方程，推导Ehrenfest-Reynolds数稳定性判据，数值验证误差增长，为初始高斯分布宽度选择提供指导。
In 1931 Koopman and von Neumann extended previous work of Liouville and provided an operatorial version of Classical Mechanics (CM). In this talk we will review a path-integral formulation of this operatorial version of CM. In particular we will study the geometrical nature of the many auxiliary variables present and of the unexpected universal symmetries generated by the functional technique.
Quantum algorithm for the advection-diffusion equation and the Koopman-von Neumann approach to nonlinear dynamical systems
本研究提出基于哈密顿量模拟线性组合技术的量子算法，结合迎风格式离散化避免虚假振荡，利用QSP、QSVT及定制块编码实现，经仿真验证复杂度与成功率，可通用模拟多类线性及非线性微分方程。
Unitary discretization of the Koopman-von Neumann equation for quantum simulation of fluid and plasma dynamics
本文提出了一种基于Weyl有序KvN生成器和求和-按部分离散化的方案，用于在量子计算机上实现流体和等离子体动力学的精确模拟，通过证明离散幺正性来克服网格分辨率的限制。
We study the time evolution of the entangled kaon system by considering the Liouville - von Neumann equation with an additional term which allows for decoherence. We choose as generators of decoherence the projectors to the 2-particle eigenstates of the Hamiltonian. Then we compare this model with the data of the CPLEAR experiment and find in this way an upper bound on the strength $λ$ of the decoherence. We also relate $λ$ to an effective decoherence parameter $ζ$ considered previously in literature. Finally we discuss our model in the light of different measures of entanglement, i.e. the von…
In this paper we examine in detail the non-commutative symplectic algebra underlying quantum dynamics. We show that this algebra contains both the Weyl-von Neumann algebra and the Moyal algebra. The latter contains the Wigner distribution as the kernel of the density matrix. The underlying non-commutative geometry can be projected into either of two Abelian spaces, so-called `shadow phase spaces'. One of these is the phase space of Bohmian mechanics, showing that it is a fragment of the basic underlying algebra. The algebraic approach is much richer, giving rise to two fundamental dynamical ti…
Time complexity analysis of quantum algorithms via linear representations for nonlinear ordinary and partial differential equations
本研究构建基于线性表示的非线性ODEs与HJE量子算法，对比不同离散化下的两类量子方法时间复杂度，提出求解HJE的Schrödinger框架并与Liouville框架对比。
We use the Liouville-von Neumann (LvN) approach to study the dynamics and the adiabaticity of a time-dependent driven anharmonic oscillator as an eample of non-equilibrium quantum dynamics. We show that the adiabaticity is minimally broken in the sense that a gaussian wave-packet at the past infinity evolves to coherent states, however slowly the potential changes, its coherence factor is order of the coupling. We also show that the dynamics are governed by an equation of motion similar to the Kepler motion which satisfies the angular momentum conservation.
The Infinitesimal Structure of Quantum Information
该研究为量子态空间建立严格统一几何框架，通过嵌入高阶对偶数代数，将量子态表示为非约化概型理论点，使控制刘维尔 - 冯·诺依曼动力学的换位子映射到代数流上，确立幂零对偶代数为高维量子运动学几何图景。
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