一个粗粒度的维格纳分布p_{W}（x，u）服从正性推导出的信息论的考虑。设p（x，u）为纯态粒子位置和动量涨落x，u的未知联合概率密度函数。假设其傅里叶变换F. T. [p(x，u）]=| Z（x，z）|exp[iPsi（x，z）]被构造为全息图。(Such全息图通常用于外差干涉测量法。考虑一个粒子随机地照亮这个相位全息图。测量其两个位置坐标。要求测量值通过相位函数Psi（x，z）的变化包含有关真实位置的极大量Fisher信息。极值解给出输出PDF p（x，u），其是Wigner p_{W}（x，u）与定义位置x或动量u中的不确定性的仪器函数的卷积。卷积自然地从该方法中产生，并且是一维的，与通常用于粗粒度目的的二维卷积相比。如果一维工具函数足够宽，则输出服从正性，如PDF所要求的那样。这一结果适用于一大类系统：那些振幅a（x）在其边界处相同的系统（例如：具有正宇称的状态a（x）;具有周期性边界条件;被困在盒子中的自由粒子）。
英文摘要：
A coarse grained Wigner distribution p_{W}(x,u) obeying positivity derives out of information-theoretic considerations. Let p(x,u) be the unknown joint PDF (probability density function) on position- and momentum fluctuations x,u for a pure state particle. Suppose that the phase part Psi(x,z) of its Fourier transform F.T.[p(x,u)]=|Z(x,z)|exp[iPsi(x,z)] is constructed as a hologram. (Such a hologram is often used in heterodyne interferometry.) Consider a particle randomly illuminating this phase hologram. Let its two position coordinates be measured. Require that the measurements contain an extreme amount of Fisher information about true position, through variation of the phase function Psi(x,z). The extremum solution gives an output PDF p(x,u) that is the convolution of the Wigner p_{W}(x,u) with an instrument function defining uncertainty in either position x or momentum u. The convolution arises naturally out of the approach, and is one-dimensional, in comparison with the two-dimensional convolutions usually proposed for coarse graining purposes. The output obeys positivity, as required of a PDF, if the one-dimensional instrument function is sufficiently wide. The result holds for a large class of systems: those whose amplitudes a(x) are the same at their boundaries (Examples: states a(x) with positive parity; with periodic boundary conditions; free particle trapped in a box).
On the significance of Wigner's Friend in contexts beyond quantum foundations
Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences（奥地利科学院量子光学与量子信息研究所）；Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna（维也纳大学物理学院量子科学与技术维也纳中心）；Perimeter Institute for Theoretical Physics（理论物理珀米特研究所）
We present the experimental reconstruction of the Wigner function of an individual electronic spin qubit associated with a nitrogen-vacancy (NV) center in diamond at room temperature. This spherical Wigner function contains the same information as the density matrix for arbitrary spin systems. As an example, we exactly characterize the quantum state of a single qubit undergoing a nearly pure dephasing process by Wigner function. The fidelities and purities during this process are extracted from the experimental reconstructed Wigner functions, whose dynamics agree with the theoretical predictio…
The significance of the de Broglie-Bohm hidden-particle position in the relativistic regime is addressed, seeking connection to the (orthodox) single-particle Newton-Wigner position. The effect of non-positive excursions of the ensemble density for extreme cases of positive-energy waves is easily computed using an integral of the equations of motion developed here for free spin-0 particles in 1+1 dimensions and is interpreted in terms of virtual-like pair creation and annihilation beneath the Compton wavelength. A Bohmtheoretic description of the acausal explosion of a specific Newton-Wigner-l…
We examine the effect of the decoherence-induced reduction of correlation length on a one-dimensional scattering problem by solving numerically the evolution equation for the Wigner function with decoherence proposed in [L. Barletti, G. Frosali and E. Giovannini, Journal of Computational and Theoretical Transport 47, 209 (2018)]. The numerical solution is achieved by the Splitting-Scheme algorithm. Three cases are examined, corresponding to a reflection-dominated regime, a transmission-dominated regime and an intermediate one. The dynamic evolution of the Wigner function is followed until the …
Wigner entropy conjecture and the interference formula in quantum phase space
Wyant College of Optical Sciences, The University of Arizona（亚利桑那大学怀特光学科学学院）；DIENS, École Normale Supérieure, PSL University, CNRS, INRIA (QAT)（巴黎高等师范学院，PSL大学，法国国家科学研究中心，INRIA（QAT））
In this paper, the ground state Wigner function of a many-body system is explored theoretically and numerically. First, an eigenvalue problem for Wigner function is derived based on the energy operator of the system. The validity of finding the ground state through solving this eigenvalue problem is obtained by building a correspondence between its solution and the solution of stationary Schrödinger equation. Then, a numerical method is designed for solving proposed eigenvalue problem in one dimensional case, which can be briefly described by i) a simplified model is derived based on a quantum…
We assess whether a confined Wigner molecule constituted by $2N$ fermions behaves as $N$ bosons or $2N$ fermions. Following the work by C. K. Law [Phys. Rev. A \textbf{71}, 034306 (2005)] and Chudzicki et al. [Phys. Rev. Lett. \textbf{104}, 070402 (2010)] we discuss the physical meaning and the reason why a large amount of entanglement is needed in order to ensure a bosonic composite behavior. By applying a composite boson ansatz, we found that a Wigner molecule confined in two dimensional traps presents a bosonic behavior induced by symmetry. The two-particle Wigner molecule ground state requ…
Phase-space features of the Wigner flow for generic one-dimensional systems with a Hamiltonian, $H^{W}(q,\,p)$, constrained by the $\partial ^2 H^{W} / \partial q \partial p = 0$ condition are analytically obtained in terms of Wigner functions and Wigner currents. Liouvillian and stationary profiles are identified for thermodynamic (TD) and Gaussian quantum ensembles to account for exact corrections due to quantum modifications over a classical phase-space pattern. General results are then specialized to the Harper Hamiltonian system which, besides working as a feasible test platform for the f…
By means of a well-grounded mapping scheme linking Schwinger unitary operators and generators of the special unitary group $\mathrm{SU(N)}$, it is possible to establish a self-consistent theoretical framework for finite-dimensional discrete phase spaces which has the discrete $\mathrm{SU(N)}$ Wigner function as a legitimate by-product. In this paper, we apply these results with the aim of putting forth a detailed study on the discrete $\mathrm{SU(2)} \otimes \mathrm{SU(2)}$ and $\mathrm{SU(4)}$ Wigner functions, in straight connection with experiments involving, among other things, the tomogra…
Quantum coherence is a crucial resource for quantum information processing. By employing the language of coherence orders largely applied in NMR systems, quantum coherence has been currently addressed in terms of multiple quantum coherences (MQCs). Here we investigate the $α$-MQCs, a novel class of multiple quantum coherences which is based on $α$-relative purity, an information-theoretic quantifier analogous to quantum fidelity and closely related to Rényi relative entropy of order $α$. Our framework enables linking $α$-MQCs to Wigner-Yanase-Dyson skew information (WYDSI), an asymmetry monoto…
We present a self-consistent theoretical framework for finite-dimensional discrete phase spaces that leads us to establish a well-grounded mapping scheme between Schwinger unitary operators and generators of the special unitary group $\mathrm{SU(N)}$. This general mathematical construction provides a sound pathway to the formulation of a genuinely discrete Wigner function for arbitrary quantum systems described by finite-dimensional state vector spaces. To illustrate our results, we obtain a general discrete Wigner function for the group $\mathrm{SU(3)}$ and apply this to the study of a partic…
The stochastic solutions to the Wigner equation, which explain the nonlocal oscillatory integral operator $Θ_V$ with an anti-symmetric kernel as {the generator of two branches of jump processes}, are analyzed. All existing branching random walk solutions are formulated based on the Hahn-Jordan decomposition $Θ_V=Θ^+_V-Θ^-_V$, i.e., treating $Θ_V$ as the difference of two positive operators $Θ^\pm_V$, each of which characterizes the transition of states for one branch of particles. Despite the fact that the first moments of such models solve the Wigner equation, we prove that the bounds of corr…
We present an extensive experimental study of the distributions of the real and imaginary parts of the off-diagonal elements of the scattering matrix $\hat S$ and the Wigner's reaction $\hat K$-matrix for open microwave networks with broken time ($T$) reversal invariance. Microwave Faraday circulators were applied in order to break $T$-invariance. The experimental distributions of the real and imaginary parts of the off-diagonal entries of the scattering matrix $\hat S$ are compared with the theoretical predictions from the supersymmetry random matrix theory [A. Nock, S. Kumar, H.-J. Sommers, …
Wave-function microscopy: Derivation and anatomy of exact algebraic spinful wave functions and full Wigner-molecular spectra of a few highly correlated rapidly rotating ultracold fermionic atoms
To implement the Wigner branching random walk, the particle carrying a signed weight, either $-1$ or $+1$, is more friendly to data storage and arithmetic manipulations than that taking a real-valued weight continuously from $-1$ to $+1$. The former is called a signed particle and the latter a weighted particle. In this paper, we propose two efficient strategies to realize the signed-particle implementation. One is to interpret the multiplicative functional as the probability to generate pairs of particles instead of the incremental weight, and the other is to utilize a bootstrap filter to adj…
Quantum decoherence is the disappearance of simple phase relations within a discrete quantum system as a result of interactions with an environment. For many applications, the question is not necessarily how to avoid (inevitable) system-environment interactions, but rather how to design environments that optimally preserve a system's phase relations in spite of such interactions. The formation of system-environment entanglement is a major driving mechanism for decoherence, and a detailed understanding of this process could inform strategies for conserving coherence optimally. This requires sca…
Probing information theoretic measures of nonlinear ultracold quantum gases using phase-space distributions
Sardar Vallabhbhai National Institute of Technology（萨达尔·瓦拉巴伊国家理工学院）；North Carolina State University（北卡罗来纳州立大学）；Indian Institute of Technology Bombay（孟买印度理工学院）
Many practical sampling patterns for function approximation on the rotation group utilizes regular samples on the parameter axes. In this paper, we relate the mutual coherence analysis for sensing matrices that correspond to a class of regular patterns to angular momentum analysis in quantum mechanics and provide simple lower bounds for it. The products of Wigner d-functions, which appear in coherence analysis, arise in angular momentum analysis in quantum mechanics. We first represent the product as a linear combination of a single Wigner d-function and angular momentum coefficients, otherwis…
