Peter Schupp
We review Wehrl's definition of a semiclassical entropy in terms of coherent states and give an introductory overview of Lieb's conjecture, its proof (including earlier results), generalizations, and the role of covariant quantum channels in this context. These structures motivate an alternative definition of coherent states and have interesting physical applications and implications.
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The linear entropy, the von Neumann entropy and the atomic Wehrl entropy are frequently used to quantify entanglement in the quantum systems. These relations provide typical information on the entanglement in the Jaynes-Cummings model (JCM). In this Letter, we explain the origin of this analytically and derive a closed form for the atomic Wehrl entropy. Moreover, we show that it is more convenient to use the Bloch sphere radius for quantifying entanglement in the JCM instead of these entropic relations.
The Wehrl entropy is an entropy associated to the Husimi quasi-probability distribution. We discuss how it can be used to formulate entropic uncertainty relations and for a quantification of entanglement in continuous variables. We show that the Wehrl-Lieb inequality is closer to equality than the usual Białynicki-Birula and Mycielski entropic uncertainty relation almost everywhere. Furthermore, we show how a Wehrl mutual information can be used to obtain a measurable perfect witness for pure state bipartite entanglement, which additionally provides a lower bound on the entanglement entropy.
The Wehrl information entropy and its phase density, the so-called Wehrl phase distribution, are applied to describe Schrödinger cat and cat-like (kitten) states. The advantages of the Wehrl phase distribution over the Wehrl entropy in a description of the superposition principle are presented. The entropic measures are compared with a conventional phase distribution from the Husimi Q-function. Compact-form formulae for the entropic measures are found for superpositions of well-separated states. Examples of Schrödinger cats (including even, odd and Yurke-Stoler coherent states), as well as the…
We propose to quantify the entanglement of pure states of $N \times N$ bipartite quantum system by defining its Husimi distribution with respect to $SU(N)\times SU(N)$ coherent states. The Wehrl entropy is minimal if and only if the pure state analyzed is separable. The excess of the Wehrl entropy is shown to be equal to the subentropy of the mixed state obtained by partial trace of the bipartite pure state. This quantity, as well as the generalized (R{é}nyi) subentropies, are proved to be Schur--convex, so they are entanglement monotones and may be used as alternative measures of entanglement…
Bayesian Interpretation of Husimi Function and Wehrl Entropy
本文借助贝叶斯定理，为自旋系统中的Husimi函数和Wehrl熵提供了一种基于直接测量的概率解释，揭示了其与经典陀螺相空间分布的对应关系，深化了对经典-量子对应关系的理解。
In this manuscript we study the Wehrl entropy of entangled oscillators. This semiclassical entropy associated with the phase-space description of quantum mechanics can be used for formulating uncertainty relations and for a quantification of entanglement. We focus on a system of two coupled oscillators described within its Segal-Bargmann space. This Hilbert space of holomorphic functions integrable with respect to a given Gaussian-like measure is particularly convenient to deal with harmonic oscillators. Indeed, the Stone-von Neumann theorem allows us to work in this space in a full correspond…
We propose a method to identify the order of a Quantum Phase Transition by using area measures of the ground state in phase space. We illustrate our proposal by analyzing the well known example of the Quantum Cusp, and four different paradigmatic boson models: Dicke, Lipkin-Meshkov-Glick, interacting boson model, and vibron model.
We introduce a new class of unitary transformations based on the su(1,1) Lie algebra that generalizes, for certain particular representations of its generators, well-known squeezing transformations in quantum optics. To illustrate our results, we focus on the two-mode bosonic representation and show how the parametric amplifier model can be modified in order to generate such a generalized squeezing operator. Furthermore, we obtain a general expression for the bipartite Wigner function which allows us to identify two distinct sources of entanglement, here labelled by dynamical and kinematical e…
The minimum Renyi and Wehrl output entropies are found for bosonic channels in which the signal photons are either randomly displaced by a Gaussian distribution (classical-noise channel), or in which they are coupled to a thermal environment through lossy propagation (thermal-noise channel). It is shown that the Renyi output entropies of integer orders z>1 and the output Wehrl entropy are minimized when the channel input is a coherent state.
We propose a new method to identify transitions from a topological insulator to a band insulator in silicene (the silicon equivalent of graphene) in the presence of perpendicular magnetic and electric fields, by using the Rényi-Wehrl entropy of the quantum state in phase space. Electron-hole entropies display an inversion/crossing behavior at the charge neutrality point for any Landau level, and the combined entropy of particles plus holes turns out to be maximum at this critical point. The result is interpreted in terms of delocalization of the quantum state in phase space. The entropic descr…
Dynamics of a periodically time dependent quantum system is reflected in the features of the eigenstates of the Floquet operator. Of the special importance are their localization properties quantitatively characterized by the eigenvector entropy, the inverse participation ratio or the eigenvector statistics. Since these quantities depend on the choice of the eigenbasis, we suggest to use the overcomplete basis of coherent states, uniquely determined by the classical phase space. In this way we define the mean Wehrl entropy of eigenvectors of the Floquet operator and demonstrate that this quant…
We show that the effective coarse graining of a two-mode squeezed density matrix, implicit in the Wehrl approaches to a semiclassical phase-space distribution, leads to results in agreement with previous different definitions of entropy for the process of pair production from the vacuum. We also present, in this context, a possible interpretation of the entropy growth as an amplification (due to the squeezing) of our lack of knowledge about the initial conditions, which gives rise to an effective decoherence of the squeezed density matrix.
The quench dynamics of many-body quantum systems may exhibit non-analyticities in the Loschmidt echo, a phenomenon known as dynamical phase transition (DPT). Despite considerable research into the underlying mechanisms behind this phenomenon, several open questions still remain. Motivated by this, we put forth a detailed study of DPTs from the perspective of quantum phase space and entropy production, a key concept in thermodynamics. We focus on the Lipkin-Meshkov-Glick model and use spin coherent states to construct the corresponding Husimi-$Q$ quasi-probability distribution. The entropy of t…
We prove two new fundamental uncertainty relations with quantum memory for the Wehrl entropy. The first relation applies to the bipartite memory scenario. It determines the minimum conditional Wehrl entropy among all the quantum states with a given conditional von Neumann entropy and proves that this minimum is asymptotically achieved by a suitable sequence of quantum Gaussian states. The second relation applies to the tripartite memory scenario. It determines the minimum of the sum of the Wehrl entropy of a quantum state conditioned on the first memory quantum system with the Wehrl entropy of…
An elementary approach to Wehrl-type entropy bounds in quantitative form
本文通过基本方法证明了对称SU(N)相干态的Lieb-Solovej不等式稳定性，利用了Wehrl型熵在复数空间单位球上的重新表述及显式计算。
We study the Husimi distribution of the ground state in the Dicke model of field-matter interactions to visualize the quantum phase transition, from normal to superradiant, in phase-space. We follow an exact numerical and variational analysis, without making use of the usual Holstein-Primakoff approximation. We find that Wehrl entropy of the Husimi distribution provides an indicator of the sharp change of symmetry trough the critical point. Additionally, we note that the zeros of the Husimi distribution characterize the Dicke model quantum phase transition.
We prove in the multimode scenario a fundamental relation between the Wehrl and the von Neumann entropy, stating that the minimum Wehrl entropy among all the quantum states with a given von Neumann entropy is achieved by thermal Gaussian states. We also prove that thermal Gaussian input states minimize the output von Neumann entropy of multimode quantum Gaussian attenuators, amplifiers and phase-contravariant channels among all the input states diagonal in some product basis and with a given entropy. This result constitutes a major step towards the proof of the same property for generic input …
Gnutzmann and Zyczkowski have proposed the Renyi-Wehrl entropy as a generalization of the Wehrl entropy, and conjectured that its minimum is obtained for coherent states. We prove this conjecture for the Renyi index q=2,3,... in the cases of compact semisimple Lie groups. A general formula for the minimum value is given.
We determine the minimum Wehrl entropy among the quantum states with a given von Neumann entropy, and prove that it is achieved by thermal Gaussian states. This result determines the relation between the von Neumann and the Wehrl entropies. The key idea is proving that the quantum-classical channel that associates to a quantum state its Husimi Q representation is asymptotically equivalent to the Gaussian quantum-limited amplifier with infinite amplification parameter. This equivalence also permits to determine the p->q norms of the aforementioned quantum-classical channel in the two particular…
Nonclassical states of a quantized light are described in terms of Glauber-Sudarshan P distribution which is not a genuine classical probability distribution. Despite several attempts, defining a uniform measure of nonclassicality (NC) for the single mode quantum states of light is yet an open task. In our previous work [Phys. Rev. A 95, 012330 (2017)] we have shown that the existing well-known measures fail to quantify the NC of single mode states that are generated under multiple NC-inducing operations. Recently, Ivan et. al. [Quantum. Inf. Process. 11, 853 (2012)] have defined a measure of …
We discuss thermalization of isolated quantum systems by using the Husimi-Wehrl entropy evaluated in the semiclassical treatment. The Husimi-Wehrl entropy is the Wehrl entropy obtained by using the Husimi function for the phase space distribution. The time evolution of the Husimi function is given by smearing the Wigner function, whose time evolution is obtained in the semiclassical approximation. We show the efficiency and usefullness of this semiclassical treatment in describing entropy production of a couple of quantum mechanical systems, whose classical counter systems are known to be chao…
The Wehrl-type entropy conjecture for symmetric $SU(N)$ coherent states: cases of equality and stability
本文证明对称 $SU(N)$ 表示的 Wehrl 型熵仅由对称相干态最小化，解决 $N\geq3$ 时的唯一性问题，并给出全纯多项式集中性、Faber–Krahn 不等式及尖锐定量稳定性界。
Quantifying complexity of continuous-variable quantum states via Wehrl entropy and Fisher information
本文提出基于Wehrl熵和Fisher信息量化连续变量量子态复杂性的方法，并通过高斯和非高斯态验证其有效性。
We generalize the concept of the Wehrl entropy of quantum states which gives a basis-independent measure of their localization in phase space. We discuss the minimal values and the typical values of these R{enyi-Wehrl entropies for pure states for spin systems. According to Lieb's conjecture the minimal values are provided by the spin coherent states. Though Lieb's conjecture remains unproven, we give new proofs of partial results that may be generalized for other systems. We also investigate random pure states and calculate the mean Renyi-Wehrl entropies averaged over the natural measure in t…
Wehrl Entropy and Entanglement Complexity of Quantum Spin Systems
研究N个自旋1/2粒子系统中Wehrl熵与多体量子纠缠的关系，发现每粒子Wehrl熵（WEPP）能定量反映纠缠复杂度且无需系统分割，并据此将多体纯纠缠态分为三类。
Wehrl used Glauber coherent states to define a map from quantum density matrices to classical phase space densities and conjectured that for Glauber coherent states the mininimum classical entropy would occur for density matrices equal to projectors onto coherent states. This was proved by Lieb in 1978 who also extended the conjecture to Bloch SU(2) spin-coherent states for every angular momentum $J$. This conjecture is proved here. We also recall our 1991 extension of the Wehrl map to a quantum channel from $J$ to $K=J+1/2, J+1, ...$, with $K=\infty$ corresponding to the Wehrl map to classica…
The Husimi distribution is proposed for a phase space analysis of quantum phase transitions in the two-dimensional $U(3)$ vibron model for $N$-size molecules. We show that the inverse participation ratio and Wehrl's entropy of the Husimi distribution give sharp signatures of the quantum (shape) phase transition from linear to bent. Numerical results are complemented with a variational approach using parity-symmetry-adapted $U(3)$ coherent states, which reach the minimum Wehrl entropy $\frac{N(3+2N)}{(N+1)(N+2)}$, in the rigidly linear phase, according to a generalized Wehrl-Lieb conjecture. We…
Estimation of the geometric measure of entanglement with Wehrl Moments through Artificial Neural Networks
研究利用人工神经网络基于有限的Wehrl矩输入准确预测对称多量子比特纯态和混合态的纠缠几何度量，发现ANN优于收敛加速算法，并提供实验协议，为实验估计纠缠度量提供新视角。
Exact Relation Between Wehrl-Rényi Entropy and Many-Body Entanglement
本研究建立了Wehrl-Rényi熵与多体纠缠之间的精确关系，并提出实验测量方案，用于分析强关联自旋系统。
The wave function of stabilizer states and the Wehrl conjecture
本文针对局部紧Abel群上的量子系统，完整刻画了稳定子态的波函数描述，证明其恰为Wehrl熵泛函的极小化子，并构造了稳定子态的模空间及有限情形下的计数公式。
We propose several pseudoentropy measures that agree well with the Wehrl entropy, but are significantly faster to compute. All of them are rotationally invariant measures of entanglement very sensitive to non-Gaussianity, anisotropy, and statistical dependence of spherical harmonic coefficients. We provide a simple proof that the projection pseudoentropy converges to the Wehrl entropy with increasing dimensionality of the ancilla projection space. Furthermore, for $l=2$, we show that both the Wehrl entropy and the angular pseudoentropy can be expressed as functions of the squared chordal dista…
Quantum Rényi Entropy Functionals for Bosonic Gaussian Systems
本文提出了阶数p>1且幂κ的量子Rényi熵功率不等式，通过量子卷积混合操作在玻色高斯系统上利用Wehrl-p熵功率不等式进行推导，为量子信道容量计算和高斯纠缠见证提供了理论基础。
We propose a broadly applicable formalism for the description of coarse grained entropy production in quantum mechanical processes. Our formalism is based on the Husimi transform of the quantum state, which encodes the notion that information about any quantum state is limited by the experimental resolution. We show in two analytically tractable cases (the decay of an unstable vacuum state and reheating after cosmic inflation) that the growth rate of the Wehrl entropy associated with the Husimi function approaches the classical Kolmogorov-Sinai entropy. We also discuss various possible applica…
Orthonormal bases of extreme quantumness
本研究提出基于平均反相干度与Wehrl熵的自旋态正交归一基量子性度量，识别出极端量子性的正交测量基，揭示其对称性并对应多体系统对称子空间的极值纠缠基。
We define the Wigner entropy of a quantum state as the differential Shannon entropy of the Wigner function of the state. This quantity is properly defined only for states that possess a positive Wigner function, which we name Wigner-positive states, but we argue that it is a proper measure of quantum uncertainty in phase space. It is invariant under symplectic transformations (displacements, rotations, and squeezing) and we conjecture that it is lower bounded by $\lnπ+1$ within the convex set of Wigner-positive states. It reaches this lower bound for Gaussian pure states, which are natural min…
We analyze mathematical and physical properties of a previously introduced [J. Phys. A47, 115302 (2014)] family of $U(4)$ coherent states (CS). They constitute a matrix version of standard spin $U(2)$ CS when we add an extra (pseudospin) dichotomous degree of freedom: layer, sublattice, two-well, nucleon, etc. Applications to bilayer quantum Hall systems at fractions of filling factor $ν=2$ are discussed, where Haldane's sphere picture is generalized to a Grassmannian picture. We also extend Wehrl's definition of entropy from Glauber to Grassmannian CS and state a conjecture on the entropy low…
We study information measures in quantu mechanics, with particular emphasis on providing a quantification of the notions of classicality and predictability. Our primary tool is the Shannon - Wehrl entropy I. We give a precise criterion for phase space classicality and argue that in view of this a) I provides a measure of the degree of deviation from classicality for closed system b) I - S (S the von Neumann entropy) plays the same role in open systems We examine particular examples in non-relativistic quantum mechanics. Finally, (this being one of our main motivations) we comment on field clas…
For a simple model we derive analytic expressions of entropy exchange and coherent information, from which relations between them and the concurrence are drawn. We find that in the quantum evolution the entropy exchange exhibits behavior \textsl{opposite} to that of the concurrence, whereas the coherent information shows features very similar to those of the concurrence. The meaning of this result for general systems is discussed.
Entropy and temperature of a system in a coherent state are naturally defined on a base of a density matrix of the system. As an example, entropy and temperature are evaluated for coherent states of a harmonic oscillator and quantum field described by the Klein--Gordon--Fock equation with a source term. It is shown, in particular, that the temperature of the coherent oscillator in a ground state coincides with the effective temperature of a harmonic oscillator being in contact with a heat bath (Bloch formula) when the bath temperature tends to zero. The Bekenstein--Hawking entropy of a black h…
Quantum energy teleportation in phase space quantum mechanics
该研究在相空间量子力学中分析量子能量隐形传态协议，借助 Wigner 分布、星积及 Husimi Q 函数揭示传输能量与基态纠缠的关系，并利用 Wehrl 熵量化协议中的相干性和纠缠消耗。
We investigate thermalization process in relativistic heavy ion collisions in terms of the Husimi-Wehrl (HW) entropy defined with the Husimi function, a quantum distribution function in a phase space. We calculate the semiclassical time evolution of the HW entropy in Yang-Mills field theory with the phenomenological initial field configuration known as the McLerran-Venugopalan model in a non-expanding geometry, which has instabilty triggered by initial field fluctuations. HW-entropy production implies the thermalization of the system and it reflects the underlying dynamics such as chaoticity a…
Understanding the thermalization process in a pure quantum system is a challenge in theoretical physics. In this work, we explore possible thermalization mechanism in Yang-Mills(YM) theory by using a positive semi-definite quantum distribution function called a Husimi function which is given by a coarse graining of the Wigner function within the minimum uncertainty. Then entropy is defined in terms of the Husimi function, which is called the Husimi-Wehrl(HW) entropy. We propose two numerical methods to calculate the HW entropy. We find that it is feasible to apply the semi-classical approximat…
Coherent feedback-enhanced asymmetry of thermal process in open quantum systems: Cavity optomechanics
本研究通过相干反馈增强开放量子系统中热过程的非对称性，揭示了不可逆性与量子关联的联合分析重要性，并展示了在腔量子电动力学系统中熵产生增强的可能性。
The manuscript investigates the entropy squeezing of a qubit (two-level atom) interacting with the cavity field in the Schrödinger cat states. The effects of the Schrödinger cat states of the field on the squeezing are examined. Our results suggest that entropy squeezing is not sensitive to the superposition features of two coherent states of the Schrödinger cat states. Furthermore, it is found that entropy squeezing occurs as the qubit initially close to its maximally coherent state, and the qubit initially in the maximally coherent state can generate optimal squeezing. This also demonstrates…
Closed Quantum Boltzmann Bridges: Coherent Revivals, Hidden Microstates, and the Emergence of Classical Two-Time Entropy Conditioning
本文通过宏观子空间投影和幺正演化构建封闭量子玻尔兹曼桥模型，发现隐藏微观态可抑制相干振荡，使熵轮廓趋近经典双时条件行为。
Relative entropy measure quantifying coherence, a key property of quantum system, is proposed recently. In this note, we investigate the maximally coherent state (MCS) with respect to relative entropy measure. %(denoted by $\mathcal C_{RE}$)%. We show that there are not mixed maximally coherent states and give a complete characterization of pure maximally coherent states. Based on this characterization, for a bipartite MCS with $d_A=d_B$, we obtain that the super-additivity equality of relative entropy measure holds if and only if the state is a product state of its reduced states. From the vi…
The von Neumann entropy at the output of a bosonic channel with thermal noise is analyzed. Coherent-state inputs are conjectured to minimize this output entropy. Physical and mathematical evidence in support of the conjecture is provided. A stronger conjecture--that output states resulting from coherent-state inputs majorize the output states from other inputs--is also discussed.
We prove the conditional Entropy Power Inequality for Gaussian quantum systems. This fundamental inequality determines the minimum quantum conditional von Neumann entropy of the output of the beam-splitter or of the squeezing among all the input states where the two inputs are conditionally independent given the memory and have given quantum conditional entropies. We also prove that, for any couple of values of the quantum conditional entropies of the two inputs, the minimum of the quantum conditional entropy of the output given by the conditional Entropy Power Inequality is asymptotically ach…
Control of the von Neumann Entropy for an Open Two-Qubit System Using Coherent and Incoherent Drives
针对含相干与非相干控制、退相干率时变的开放两量子比特系统，本文采用梯度投影法和遗传算法，实现熵的极值化、目标跟踪、约束满足与恒定保持。
Coherence measures and optimal conversion for coherent states
该研究提出构建相干性度量的通用策略，涵盖纯态相对熵等多类度量，并给出非相干操作下相干态的最优转换方法，为单拷贝纯态相干性研究提供了见解。
We study the fundamental properties of the quantum f-relative entropy, where f(.) is an operator convex function. We give the equality conditions under monotonicity and joint convexity, and these conditions are more general than, since they hold for a class of operator convex functions, and different for f(t) = -ln(t) from, the previously known conditions. The quantum f-entropy is defined in terms of the quantum f-relative entropy and we study its properties giving the equality conditions in some cases. We then show that the f-generalizations of the Holevo information, the entanglement-assiste…
Entropy Flow of a Laser Beam
研究揭示了理想激光束的熵流特性，通过相位扩散率和粒子流计算熵流，对比了热束与激光束的差异。
The Holevo quantity provides an upper bound for the mutual information between the sender of a classical message encoded in quantum carriers and the receiver. Applying the strong sub-additivity of entropy we prove that the Holevo quantity associated with an initial state and a given quantum operation represented in its Kraus form is not larger than the exchange entropy. This implies upper bounds for the coherent information and for the quantum Jensen--Shannon divergence. Restricting our attention to classical information we bound the transmission distance between any two probability distributi…
A quantum coordinate-entropy formulated in quantum phase space has been recently proposed together with an entropy law that asserts that such entropy can not decrease over time. The coordinate-entropy is dimensionless, a relativistic scalar, and it is invariant under coordinate and CPT transformations. We study here the time evolution of this entropy. We show that the entropy associated with coherent states evolving under a Dirac Hamiltonian is increasing. However, for the collisions of two particles, where each is evolving as a coherent state, as they come closer to each other their spatial e…
Qumode-Based Quantum Image Storage with Entropy-Guided Frame Indexing and Fidelity-Preserved Retrieval
本文提出一种基于连续变量光子系统与qumode的量子图像存储框架，利用delta演化机制和冯·诺依曼熵引导帧索引实现高效存储与可扩展检索，并通过Strawberry Fields仿真验证了保真度保持特性。
In the framework of the Lindblad theory for open quantum systems, expressions for the density operator, von Neumann entropy and effective temperature of the damped harmonic oscillator are obtained. The entropy for a state characterized by a Wigner distribution function which is Gaussian in form is found to depend only on the variance of the distribution function. We give a series of inequalities, relating uncertainty to von Neumann entropy and linear entropy. We analyze the conditions for purity of states and show that for a special choice of the diffusion coefficients, the correlated coherent…
The position and momentum space information entropies, of the ground state of the Pöschl-Teller potential, are exactly evaluated and are found to satisfy the bound, obtained by Beckner, Bialynicki-Birula and Mycielski. These entropies for the first excited state, for different strengths of the potential well, are then numerically obtained. Interesting features of the entropy densities, owing their origin to the excited nature of the wave functions, are graphically demonstrated. We then compute the position space entropies of the coherent state of the Pöschl-Teller potential, which is known to …
The spinor condensate with spin states degenerated in the ground spin-space provides a unique platform for investigating the edge of quantum mechanics and statistical physics. We study the evolution of the condensate under the scattering of a coherent atom beam. The time-dependent magnetization, entanglement entropy, thermal entropy and the entropy production rate are calculated. A novel spontaneous symmetry breaking is found during the evolution, It is shown that the stationary spin distribution can be controlled by the incoming coherent spin state of the incident atom beam, therefore the ato…
Active Quantum Distillation
本文提出主动量子蒸馏协议，利用 bang-bang 控制相干动力学降低子系统冯·诺依曼熵，推导了粒子数守恒下熵下界，并在 Bose-Hubbard 模型上验证，实现高效玻色子蒸馏。
Heat, work and entropy production: the statistical distribution of such quantities are constrained by the fluctuation theorems (FT), which reveal crucial properties about the nature of non-equilibrium dynamics. In this paper we report theoretical and experimental results regarding two FT for a new quantity, named coherent energy, which is an energy form directly associated with the coherences of the quantum state. We also demonstrate that this quantity behaves as a thermodynamic arrow of time for unitary evolutions, that is, in the absence of entropy production. The experiment is implemented i…
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