C. A. Munoz Villegas, A. Chavez Chavez, S. Chumakov, Yu. Fofanov, A. B. Klimov
We analyze quasi probability distributions in discrete phase space related to the discrete Heisenberg-Weyl group. In particular, we discuss the relation between the Discrete Wigner and Q- functions.
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We show how quasiprobability distribution functions defined over $N^{2}$-dimensional discrete phase spaces can be used to treat physical systems described by a finite space of states which exhibit spin tunneling effects. This particular approach is then applied to the Lipkin-Meshkov-Glick model in order to obtain the time evolution of the discrete Husimi function, and as a by-product the energy gap for a symmetric combination of ground and first excited states. Moreover, we also show how an angle-based potential approach can be efficiently employed to explain qualitatively certain features of …
A note on the time-reversal symmetry for the quasiprobability distributions of work
本文探讨了做功准概率分布的时间反转对称性，为量子系统中的功统计与时间反转特性提供了理论注记。
An approach featuring $s$-parametrized quasiprobability distribution functions is developed for situations where a circular topology is observed. For such an approach, a suitable set of angle-angular momentum coherent states must be constructed in appropriate fashion.
We show, in a formal way, how a class of complex quasiprobability distribution functions may be introduced by using the fractional Fourier transform. This leads to the Fresnel transform of a characteristic function instead of the usual Fourier transform. We end the manuscript by showing a way in which the distribution we are introducing may be reconstructed by using atom-field interactions.
We study the quasiprobability representation of quantum light, as introduced by Glauber and Sudarshan, for the unified characterization of quantum phenomena. We begin with reviewing the past and current impact of this technique. Regularization and convolution methods are specifically considered since they are accessible in experiments. We further discuss more general quantum systems for which the concept of negative probabilities can be generalized, being highly relevant for quantum information science. For analyzing quantum superpositions, we apply recently developed approaches to visualize q…
We present a general theory of quasiprobability distributions on phase spaces of quantum systems whose dynamical symmetry groups are (finite-dimensional) Lie groups. The family of distributions on a phase space is postulated to satisfy the Stratonovich-Weyl correspondence with a generalized traciality condition. The corresponding family of the Stratonovich-Weyl kernels is constructed explicitly. In the presented theory we use the concept of the generalized coherent states, that brings physical insight into the mathematical formalism.
We study the possibility of reconstructing the quantum state of light in a cavity subject to dissipation. We pass atoms, also subject to decay, through the cavity and surprisingly show that both decays allow the measurement of $s$-parametrized quasiprobability distributions. In fact, if we consider only atomic decay, we show that the Wigner function may be reconstructed. Because these distributions contain whole information of the initial field state, it is possible to recover information after both atomic and field decays occur.
We propose a method to define quasiprobability distributions for general spin-$j$ systems of dimension $n=2j+1$, where $n$ is a prime or power of prime. The method is based on a complete set of orthonormal commuting operators related to Mutually Unbiased Bases which enable (i) a parameterisation of the density matrix and (ii) construction of measurement operators that can be physically realised. As a result we geometrically characterise the set of states for which the quasiprobability distribution is non-negative, and can be viewed as a joint distribution of classical random variables assuming…
A Wigner quasiprobability distribution of work
提出一种基于 Wigner 函数的功准概率分布，通过系统与量子测量装置相干耦合定义，可刻画功统计并揭示量子相干性与负性、干涉条纹等非经典特征的联系。
We introduce a general method for the construction of quasiprobability representations for arbitrary notions of quantum coherence. Our technique yields a nonnegative probability distribution for the decomposition of any classical state. Conversely, quantum phenomena are certified in terms of signed distributions, i.e., quasiprobabilities, and a residual component unaccessible via classical states. Our unifying method combines well-established concepts, such as phase-space distributions in quantum optics, with resources of quantumness relevant for quantum technologies. We apply our approach to …
We use the exact calculation of the quantum mechanical, temporal characteristic function $χ(η)$ and the degree of second-order coherence $g^{(2)}(τ)$ for a single-mode, degenerate parametric amplifier for a system in the Gaussian state, viz., a displaced-squeezed thermal state, to study the different criteria for nonclassicality. In particular, we contrast criteria that involve only one-time functions of the dynamical system, for instance, the quasiprobability distribution $P(β)$ of the Glauber-Sudarshan coherent or P-representation of the density of state and the Mandel $Q_{M}(τ)$ parameter, …
Relating the Glauber-Sudarshan, Wigner and Husimi quasiprobability distributions operationally through the quantum limited amplifier and attenuator channels
本文证明有限强度量子受限放大器和衰减器信道的单次组合可将Glauber-Sudarshan分布转化为Wigner分布，进而转化为Husimi分布，揭示了准概率分布间的操作联系及量子到经典的转变。
We demonstrate that the negative volume of any $s$-paramatrized quasiprobability, including the Glauber-Sudashan $P$-function, can be consistently defined and forms a continuous hierarchy of nonclassicality measures that are linear optical monotones. These measures therefore belong to an operational resource theory of nonclassicality based on linear optical operations. The negativity of the Glauber-Sudashan $P$-function in particular can be shown to have an operational interpretation as the robustness of nonclassicality. We then introduce an approximate linear optical monotone, and show that t…
Measurement incompatibility is a distinguishing property of quantum physics and an essential resource for many quantum information processing tasks. We introduce an approach to verify the joint measurability of measurements based on phase-space quasiprobability distributions. Our results therefore establish a connection between two notions of non-classicality, namely the negativity of quasiprobability distributions and measurement incompatibility. We show how our approach can be applied to the study of incompatibility-breaking channels and derive incompatibility-breaking sufficient conditions …
Kirkwood-Dirac Type Quasiprobabilities as Universal Identifiers of Nonclassical Quantum Resources
本文证明Kirkwood-Dirac型准概率分布可作为通用标识符揭示任意量子资源，通过不相容测量区分资源态与非资源态，其负性对应几何距离并关联反常弱值，揭示测量不相容性是量子优势的基础。
Quantum advantage from negativity of a work quasiprobability distribution
通过分析功的准概率分布负性，建立了量子电池充电过程中量子优势与初始量子相干性之间的简单关系。
Quantum Bayesian Inference in Quasiprobability Representations
本文推导了Petz恢复映射在任意拟概率表示下的表达式，并针对正规拟概率表示和SIC-POVMs表示给出了显式公式，揭示了经典与量子逻辑推断的核心差异在于对参考先验的操纵。
Quasiprobability distributions (QDs) in open quantum systems are investigated for $SU(2)$, spin like systems, having relevance to quantum optics and information. In this work, effect of both quantum non-demolition (QND) and dissipative open quantum systems, on the evolution of a number of spin QDs are investigated. Specifically, compact analytic expressions for the $W$, $P$, $Q$, and $F$ functions are obtained for some interesting single, two and three qubit states, undergoing general open system evolutions. Further, corresponding QDs are reported for an N qubit Dicke model and a spin-1 system…
The boundary of Kirkwood-Dirac quasiprobability
本文通过引入后量子准概率作为外边界，证明经典概率、KD 准概率和后量子准概率构成严格层级，并推导出适用于任意数量测量的非平凡界限，深化了对 KD 准概率在量子信息处理中应用的理解。
The standard approach to deriving fluctuation theorems fails to capture the effect of quantum correlation and coherence in the initial state of the system. Here we overcome this difficulty and derive heat exchange fluctuation theorem in the full quantum regime by showing that the energy exchange between two locally thermal states in the presence of initial quantum correlations is faithfully captured by a quasiprobability distribution. Its negativities, being associated with proofs of contextuality, are proxys of non-classicality. We discuss the thermodynamic interpretation of negative probabil…
It is well known that the squeezing spectrum of the field exiting a nonlinear cavity can be directly obtained from the fluctuation spectrum of normally ordered products of creation and annihilation operators of the cavity mode. In this article we show that the output field squeezing spectrum can be derived also by combining the fluctuation spectra of any pair of s-ordered products of creation and annihilation operators. The interesting result is that the spectrum obtained in this way from the linearized Langevin equations is exact, and this occurs in spite of the fact that no s-ordered quasipr…
Quasiprobability distributions with weak measurements
本文通过光子量子比特弱序列测量实验，结合相称和Margenau-Hill准概率方法，研究了量子相干性在时序测量中的作用，并展示了准分布用于量子监测的潜力。
We develop a Radon like transformation, in which $P$ quasiprobability distribution for spin 1/2 states is written in terms the tomographic probability distribution $w$.
A structure theorem for complex-valued quasiprobability representations of physical theories
本文基于Schmid等人的框架，将复值准概率表示扩展至无需保持恒等通道及无限维空间的情形，证明其可由对状态和恒等式作用的两个映射复合表示，为广义概率理论提供了统一数学框架。
Interferometry of quantum correlation functions to access quasiprobability distribution of work
本文提出借助辅助系统的干涉方案重构Kirkwood-Dirac准概率分布，并在金刚石氮-空位中心的电子-核自旋系统中实验重构功的准概率分布、分析其矩及测量不确定性。
Quantum circuits for measuring weak values, Kirkwood--Dirac quasiprobability distributions, and state spectra
本文提出了无需后选择即可测量弱值、KD准概率分布和密度矩阵谱的简单量子电路，基于Bargmann不变量提供了统一视角，并探讨了其在OTOCs和量子Fisher信息等任务中的应用及非经典性。
Effective algorithms for calculation of quasiprobability distributions of bright "banana'' states
针对克尔非线性产生的明亮“香蕉”态在光子数极大时维格纳函数直接计算代价过高的问题，本文提出了可适用于任意实际光子数和非线性值的 Husimi 与 Wigner 准概率函数快速计算算法。
Operational quasiprobability formulation of non-signaling in time for macrorealism violations
本文提出准时间无信号性概念，并据此定义宏观实在性的准概率表征，证明其蕴含时间无信号性但反之不成立，为测试宏观实在性提供更灵敏的工具。
Negative quasiprobability trajectories for Bell-diagonal states under local decoherence
该研究针对局域退相干下的贝尔对角态，解析确定负拟概率分布的特性，揭示其与经典随机描述的轨迹层面差异，为量子信息动力学提供了新的独特特征。
Exploring quasiprobability approach to quantum work in the presence of initial coherence: Advantages of the Margenau-Hill distribution
针对存在初始量子相干性时量子功涨落表征的准概率定义不统一问题，本文提出多项物理合理性要求，证明Margenau-Hill准概率是唯一满足全部要求的定义，并通过呼吸谐振子示例验证其经典极限收敛性。
We examine the notion of nonclassicality in terms of quasiprobability distributions. In particular, we do not only ask if a specific quasiprobability can be interpreted as a classical probability density, but require that characteristic features of classical electrodynamics are resembled. We show that the only quasiprobabilities which correctly describe the superposition principle of classical electromagnetic fields are the $s$-parameterized quasiprobabilities. Furthermore, the Glauber-Sudarshan P function is the only quantum-mechanical quasiprobability which is transformed at a classical atte…
A mapping between operators on the Hilbert space of $N$-dimensional quantum system and the Wigner quasiprobability distributions defined on the symplectic flag manifold is discussed. The Wigner quasiprobability distribution is constructed as a dual pairing between the density matrix and the Stratonovich-Weyl kernel. It is shown that the moduli space of the Stratonovich-Weyl kernel is given by an intersection of the coadjoint orbit space of the $SU(N)$ group and a unit $(N-2)$-dimensional sphere. The general consideration is exemplified by a detailed description of the moduli space of 2, 3 and …
We show that discrete quasiprobability distributions defined via the discrete Heisenberg-Weyl group can be obtained as discretizations of the continuous $SU(N)$ quasiprobability distributions. This is done by identifying the phase-point operators with the continuous quantisation kernels evaluated at special points of the phase space. As an application we discuss the positive-P function and show that its discretization can be used to treat the problem of diverging trajectories. We study the dissipative long-range transverse-field Ising chain and show that the long-time dynamics of local observa…
Polarization quasiprobability distribution defined in the Stokes space shares many important properties with the Wigner function for the position and momentum. Most notably, they both give correct one-dimensional marginal probability distributions and therefore represent the natural choice for the probability distributions in classical hidden-variable models. In this context, negativity of the Wigner function is considered as a proof of non-classicality for a quantum state. On the contrary, the polarization quasiprobability distribution demonstrates negativity for all quantum states. This feat…
The work is a concept of fundamental importance in thermodynamics. An open question is how to describe the work fluctuation for quantum coherent processes in the presence of initial quantum coherence in the energy basis. With the aim of giving a unified description, here we introduce and study a class of quasiprobability distributions of work, which give an average work equal to the average energy change of the system and reduce to the two-projective measurement scheme for an initial incoherent state. Moreover, we characterize the work with the help of fluctuation relations. In particular, by …
It is shown how to write the Kirkwood-Rihaczec quasiprobability distribution as an expectation value of the vacuum state. We do this, by writing the position eigenstates as a "displacment" of the vacumm. We also give a relation between the Glauber-Sudarshan and Kirkwood-Rihaczec quasiprobability distributions.
Quasiprobability distribution of work in the quantum Ising model
本文研究横场伊辛模型在全局和局域淬火下的功准概率分布，发现局域淬火可通过负四阶矩揭示量子上下文性，并分析了量子相变临界特征及初始相干性的资源作用。
In the present report we discuss measures of classicality/quantumness of states of finite-dimensional quantum systems, which are based on a deviation of quasiprobability distributions from true statistical distributions. Particularly, the dependence of the global indicator of classicality on the assigned geometry of a quantum state space is analysed for a whole family of Wigner quasiprobability representations. General considerations are exemplified by constructing the global indicator of classicality/quantumness for the Hilbert-Schmidt, Bures and Bogoliubov-Kubo-Mori ensembles of qubits and q…
This paper aims to explore the inherent connection among Heisenberg groups, quantum Fourier transform and (quasiprobability) distribution functions. Distribution functions for continuous and finite quantum systems are examined first as a semiclassical approach to quantum probability distribution. This leads to studying certain functionals of a pair of "conjugate" observables, connected via the quantum Fourier transform. The Heisenberg groups emerge naturally from this study and we take a rapid look at their representations. The quantum Fourier transform appears as the intertwining operator of …
Temporal Kirkwood-Dirac Quasiprobability Distribution and Unification of Temporal State Formalisms through Temporal Bloch Tomography
本文提出时间Kirkwood-Dirac准概率分布，通过时间Bloch成像统一时间态形式化，揭示其非经典特性。
We show that postselection offers a nonclassical advantage in metrology. In every parameter-estimation experiment, the final measurement or the postprocessing incurs some cost. Postselection can improve the rate of Fisher information (the average information learned about an unknown parameter from an experimental trial) to cost. This improvement, we show, stems from the negativity of a quasiprobability distribution, a quantum extension of a probability distribution. In a classical theory, in which all observables commute, our quasiprobability distribution can be expressed as real and nonnegati…
We propose a method for classical simulation of finite-dimensional quantum systems, based on sampling from a quasiprobability distribution, i.e., a generalized Wigner function. Our construction applies to all finite dimensions, with the most interesting case being that of qubits. For multiple qubits, we find that quantum computation by Clifford gates and Pauli measurements on magic states can be efficiently classically simulated if the quasiprobability distribution of the magic states is non-negative. This provides the so far missing qubit counterpart of the corresponding result [V. Veitch et …
Operational Quasiprobability in Quantum Thermodynamics: Work Extraction by Coherence and Non-joint Measurability
本研究将操作准概率（OQ）用作功分布，证明相干性与非联合可测性可提升量子系统的可提取功，还发现准概率负性大小并非非经典功的可靠指标。
Negative probability values have been widely employed as an indicator of the nonclassicality of quantum systems. Known as a quasiprobability distribution, they are regarded as a useful tool that provides significant insight into the underlying fundamentals of quantum theory when compared to the classical statistics. However, in this approach, an operational interpretation of these negative values with respect to the definition of probability---the relative frequency of occurred event---is missing. An alternative approach is therefore considered where the quasiprobability operationally reveals …
We show how mapping techniques inherent to $N^{2}$-dimensional discrete phase spaces can be used to treat a wide family of spin systems which exhibits squeezing and entanglement effects. This algebraic framework is then applied to the modified Lipkin-Meshkov-Glick (LMG) model in order to obtain the time evolution of certain special parameters related to the Robertson-Schrödinger (RS) uncertainty principle and some particular proposals of entanglement measure based on collective angular-momentum generators. Our results reinforce the connection between both the squeezing and entanglement effects…
It is commonly accepted that a deviation of the Wigner quasiprobability distribution of a quantum state from a proper statistical distribution signifies its nonclassicality. Following this ideology, we introduce the global indicator $\mathcal{Q}_N$ for quantification of "classicality-quantumness" correspondence in the form of the functional on the orbit space $\mathcal{O}[\mathfrak{P}_N]$ of the $SU(N)$ group adjoint action on the state space $\mathfrak{P}_N$ of an $N$-dimensional quantum system. The indicator $\mathcal{Q}_{N}$ is defined as a relative volume of a subspace $\mathcal{O}[\mathfr…
Kirkwood-Dirac classical states based on discrete Fourier transform: Representation with directed graph
本文研究了当两个正交基之间过渡矩阵为离散傅里叶变换矩阵时，Kirkwood-Dirac 经典态集合的结构特征，通过有向图定义了任意维数希尔伯特空间中的经典态，并证明了其凸包性质。
Decoherence from universal tomographic measurements
本文通过两种等效方法研究了从环境监控量子系统状态产生的退相干现象，展示了退相干如何使任意给定的 quasiprobability 分布显式正，从而在某种程度上建模经典性的出现。
Quantum Coherence and Anomalous Work Extraction in Qubit Gate Dynamics
本文提出基于Kirkwood-Dirac准概率分布的框架，量化了相干性在通用循环量子演化中对功提取的贡献，特别研究了异常过程，即在准概率分布负值的情况下，即使单个过程与能量增益相关，仍能提取功。通过分析量子门序列，建立了全电路功统计与构成门的功统计之间的组合关系，揭示了相干性在量子计算热力学中的作用。
We explain, on the example of Wigner's quasiprobability distribution, how negative probabilities may be used in the foundations of probability.
Coherent distributions for the rigid rotator
本研究提出刚性转子经典刘维尔方程的相干解为正相空间分布，经形式离散化得到维格纳型准概率分布，其结果与各向异性转子量子化一致且含有限零点能，还揭示了准概率分布演化时量子波函数需满足含时薛定谔方程。
Quantifying Margenau--Hill Nonclassicality
本文提出基于矩的框架量化Margenau--Hill准概率分布的非经典性，引入对数负性并推导下界、上界及正性阈值，支持多副本估计。
Quantum dynamics simulations can be improved using novel quasiprobability distributions based on non-orthogonal hermitian kernel operators. This introduces arbitrary functions (gauges) into the stochastic equations, which can be used to tailor them for improved calculations. A possible application to full quantum dynamic simulations of BEC's is presented.
We show that a nonlinear asymmetric directional coupler composed of a linear waveguide and a nonlinear waveguide operating by nondegenerate parametric amplification is an effective source of single-mode squeezed light. This is has been demonstrated, under certain conditions and for specific modes, for incident coherent beams in terms of the quasiprobability functions, photon-number distribution and phase distribution.
Quantum light depolarization is handled through a master equation obtained by coupling dispersively the field to a randomly distributed atomic reservoir. This master equation is solved by transforming it into a quasiprobability distribution in phase space and the quasiclassical limit is investigated.
Quantum beating and cyclic structures in the phase-space dynamics of the Kramers-Henneberger atom
该研究通过Wigner准概率分布求解含时薛定谔方程，发现KH原子在相空间中呈现与能量差相关的循环运动，且动量上界受限，并指出初始制备于KH基态时完整动力学最稳定。
We report on the first experimental reconstruction of an entanglement quasiprobability. In contrast to related techniques, the negativities in our distributions are a necessary and sufficient identifier of separability and entanglement and enable a full characterization of the quantum state. A reconstruction algorithm is developed, a polarization Bell state is prepared, and its entanglement is certified based on the reconstructed entanglement quasiprobabilities, with a high significance and without correcting for imperfections.
Revealing entanglement through local features of phase-space distributions
通过相空间分布的局部特征，提出一种检测非高斯纠缠的连续变量分离性判据，并设计简单测量方案实现高效检测。
We investigate propagation of few-photon pulses in waveguides coupled to a two-level system by means of the method of distribution functions in coordinate-momentum space that provides a detailed description of photon systems. We find that the distribution function of the transmitted pulse can be negative for the nonclassical input (i.e., single-photon Fock state). This reveals the quasiprobability nature of photon distribution. Analytical expressions for photon densities in the momentum space as well as in the coordinate space are obtained for the mentioned single-photon Gaussian input. We als…
We show that a quantum state may be represented as the sum of a joint probability and a complex quantum modification term. The joint probability and the modification term can both be observed in successive projective measurements. The complex modification term is a measure of measurement disturbance. A selective phase rotation is needed to obtain the imaginary part. This leads to a complex quasiprobability, the Kirkwood distribution. We show that the Kirkwood distribution contains full information about the state if the two observables are maximal and complementary. The Kirkwood distribution g…
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