Minoru Horibe, Takaaki Hashimoto, Akihisa Hayashi
We show that the Fano operator for one dimensional quantum system is uniquely determined by assuming the reasonable behavior under translation and parity transformation on phase space. Contrarily, for the system with lattice phase space the same procedure does not work.
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Notwithstanding radical conceptual differences between classical and quantum mechanics, it is usually assumed that physical measurements concern observables common to both theories . Not so with the eigenvalues ($\pm 1$) of the parity operator. The effect of such a measurement on a mixture of even and odd states of the harmonic oscillator is akin to separating at a single stroke a pair of shuffled card decks: the result is a set of definite parity, though otherwise mixed. The Wigner function should be a sensitive probe for this phenomenon, for it can be interpreted as the expectation value of …
This is a 'facsimile-style' translation of Wigner's seminal paper on measurement limitations in the presence of additive conservation laws. A critical survey of the history of subsequent extensions and variations of what is now known as the Wigner-Araki-Yanase (WAY) Theorem is provided in a paper published concurrently.
A finite Hilbert space can be associated to a periodic phase space, that is, a torus. A finite subgroup of operators corresponding to reflections and translations on the torus form respectively the basis for the discrete Weyl representation, including the Wigner function, and for its Fourier conjugate, the chord representation. They are invariant under Clifford transformations and obey analogous product rules to the continuous representations, so allowing for the calculation of expectations and correlations for observables. We here import new identities from the continuum for products of pure …
The Heisenberg-Weyl-parity group its coherent states and a unified Wigner-Weyl function
本文研究了Heisenberg-Weyl-parity群的结构及其相干态，探讨了该群作为广义二面体群的性质，并提出了统一的Wigner-Weyl函数及其应用。
We demonstrate that the Wigner function of the Einstein-Podolsky-Rosen state, though positive definite, provides a direct evidence of the nonlocal character of this state. The proof is based on an observation that the Wigner function describes correlations in the joint measurement of the phase space displaced parity operator.
A parity-dependent squeezing operator is introduced which imposes different SU(1,1) rotations on the even and odd subspaces of the harmonic oscillator Hilbert space. This operator is used to define parity-dependent squeezed states which exhibit highly nonclassical properties such as strong antibunching, quadrature squeezing, strong oscillations in the photon-number distribution, etc. In contrast to the usual squeezed states whose $Q$ and Wigner functions are simply Gaussians, the parity-dependent squeezed states have much more complicated $Q$ and Wigner functions that exhibit an interesting in…
We report a direct measurement of the Wigner function characterizing the quantum state of a light mode. The experimental scheme is based on the representation of the Wigner function as an expectation value of a displaced photon number parity operator. This allowed us to scan the phase space point-by-point, and obtain the complete Wigner function without using any numerical reconstruction algorithms.
We study the stationary Wigner equation on a bounded, one-dimensional spatial domain with inflow boundary conditions by using the parity decomposition in (Barletti and weifel, Trans. Theory Stat. Phys., 507--520, 2001). The decomposition reduces the half-range, two-point boundary value problem into two decoupled initial value problems of the even part and the odd part. Without using a cutoff approximation around zero velocity, we prove that the initial value problem for the even part is well-posed. For the odd part, we prove the uniqueness of the solution in the odd $L^2$-space by analyzing th…
We present an experimental realisation of the direct scheme for measuring the Wigner function of a single quantized light mode. In this method, the Wigner function is determined as the expectation value of the photon number parity operator for the phase space displaced quantum state.
We present a scheme that uses Ramsey interferometry to directly probe the Wigner function of a neutral atom confined in an optical trap. The proposed scheme relies on the well-established fact that the Wigner function at a given point $(x,p)$ in phase space is proportional to the expectation value of the parity operator relative to that point. In this work, we show that parity-even and parity-odd motional states can be mapped to two distinct internal states of the atom by using state-dependent trapping potentials. The Wigner function can thus be measured point-by-point in phase space with a si…
Realistic non-Gaussian operations scheme in parity detection based Mach-Zehnder quantum interferometry
本文理论分析了基于宇称检测的MZI在光子扣除、添加和催化等现实非高斯操作下的相位灵敏度，推导了统一Wigner函数，发现考虑成功概率后光子添加最具优势。
We construct an explicit Wigner function for N-mode squeezed state. Based on a previous observation that the Wigner function describes correlations in the joint measurement of the phase-space displaced parity operator, we investigate the non-locality of multipartite entangled state by the violation of Zukowski-Brukner N-qubit Bell inequality. We find that quantum predictions for such squeezed state violate these inequalities by an amount that grows with the number N.
Wigner distributions for quantum mechanical systems whose configuration space is a finite group of odd order are defined so that they correctly reproduce the marginals and have desirable transformation properties under left and right translations. While for the Abelian case we recover known results, though from a different perspective, for the non Abelian case, our results appear to be new.
Quantum simulation of chemical Hamiltonians enables the efficient calculation of chemical properties. Mapping is one of the essential steps in simulating fermionic systems on quantum computers. In this work, a unified framework of transformations mapping fermionic systems to qubit systems is presented, and many existing transformations, such as Jordan-Wigner, Bravyi-Kitaev, and parity transformations, are included in this framework. Based on this framework, the Multilayer Segmented Parity (MSP) transformation is proposed. The MSP transformation is a general mapping with an adjustable parameter…
In spite of their potential usefulness, Wigner functions for systems with SU(1,1) symmetry have not been explored thus far. We address this problem from a physically-motivated perspective, with an eye towards applications in modern metrology. Starting from two independent modes, and after getting rid of the irrelevant degrees of freedom, we derive in a consistent way a Wigner distribution for SU(1,1). This distribution appears as the expectation value of the displaced parity operator, which suggests a direct way to experimentally sample it. We show how this formalism works in some relevant exa…
Parity symmetry breaking of spin-$j$ coherent state superpositions in Gaussian noise channel
该研究利用Wigner函数与Wigner-Yanase偏斜信息的关联，分析自旋-j相干态叠加的宇称对称破缺，发现其与自旋j值正相关，且高斯噪声下退相干参数s越高破缺越显著。
Despite the indisputable merits of the Wigner phase-space formulation, it has not been widely explored for systems with SU(1,1) symmetry, as a simple operational definition of the Wigner function has proved elusive in this case. We capitalize on the unique properties of the parity operator, to derive in a consistent way a \emph{bona fide} SU(1,1) Wigner function that faithfully parallels the structure of its continuous-variable counterpart. We propose an optical scheme, involving a squeezer and photon-number-resolving detectors, that allows for direct point-by-point sampling of that Wigner fun…
We introduce a quasi-probability phase space distribution with two pairs of azimuthal-angular coordinates. This representation is well adapted to describe quantum systems with discrete symmetry. Quantum error correction of states encoded in continuous variables using translationally invariant states is studied as an example of application. We also propose an experimental scheme for measuring such new distribution.
The semiclassical Wigner function for a Bohr-quantized energy eigenstate is known to have a caustic along the corresponding classical closed phase space curve in the case of a single degree of freedom. Its Fourier transform, the semiclassical chord function, also has a caustic along the conjugate curve defined as the locus of diameters, i.e. the maximal chords of the original curve. If the latter is convex, so is its conjugate, resulting in a simple fold caustic. The uniform approximation through this caustic, that is here derived, describes the transition undergone by the overlap of the state…
In this paper, we use the displacement operator together with parity operation to construct the superposition of two coherent states. By transmitting this superposition from 50-50 beam splitter the two-mode qubit like ECS is generated. Moreover, we introduce a controllable method for producing qutrit like ECS using atom-field interaction in cavity QED and beam splitter. We will show that the distances of peaks of Wigner functions for reduced density matrices of two-mode ECS's are entanglement sensitive and can be a witness for entanglement. To confirm the results we use concurrence measure to …
The mean parity (the Wigner function at the origin) of excited binomial states, excited coherent states and excited thermal states in thermal channel is investigated in details. It is found that the single-photon excited binomial state and the single-photon excited coherent state exhibit certain similarity in the aspect of their mean parity in the thermal channel. We show the negative mean parity can be regarded as an indicator of nonclassicality of single-photon excitation of optical fields with a little coherence, especially for the single-photon excited thermal states.
The Quantum Ergodic Conjecture equates the Wigner function for a typical eigenstate of a classically chaotic Hamiltonian with a delta-function on the energy shell. This ensures the evaluation of classical ergodic expectations of simple observables, in agreement with Shnirelman's theorem, but this putative Wigner function violates several important requirements. Consequently, we transfer the conjecture to the Fourier transform of the Wigner function, that is, the chord function. We show that all the relevant consequences of the usual conjecture require only information contained within a small …
We prove two results about Wigner distributions. Firstly, that the Wigner transform is the only sesquilinear map ${\mathcal S}(\mathbb{R}^n) \times {\mathcal S}(\mathbb{R}^n) \to {\mathcal S}(\mathbb{R}^{2n})$ which is bounded and covariant under phase-space translations and linear symplectomorphisms. Consequently, the Wigner distributions form the only set of quasidistributions which is invariant under linear symplectic transformations. Secondly, we prove that the maximal group of (linear or non-linear) coordinate transformations that preserves the set of (pure or mixed) Wigner distributions …
Mutually unbiased bases and discrete Wigner functions are closely, but not uniquely related. Such a connection becomes more interesting when the Hilbert space has a dimension that is a power of a prime $N=d^n$, which describes a composite system of $n$ qudits. Hence, entanglement naturally enters the picture. Although our results are general, we concentrate on the simplest nontrivial example of dimension $N=8=2^3$. It is shown that the number of fundamentally different Wigner functions is severely limited if one simultaneously imposes translational covariance and that the generating operators …
Enhanced phase estimation in parity detection based Mach-Zehnder interferometer using non-Gaussian two-mode squeezed thermal input state
本文探讨了在基于宇称检测的马赫-曾德尔干涉仪中，对双模压缩热（TMST）态执行光子扣除、添加和催化等非高斯操作以增强相位估计的方法，推导了统一Wigner函数，并发现考虑成功概率时光子催化为最优操作。
Inspired by recent developments generalizing Jordan-Wigner dualities to higher dimensions, we develop a framework of such dualities using an algebraic formalism for translation-invariant Hamiltonians proposed by Haah. We prove that given a translation-invariant fermionic system with general $q$-body interactions, where $q$ is even, a local mapping preserving global fermion parity to a dual Pauli spin model exists and is unique up to a choice of basis. Furthermore, the dual spin model is constructive, and we present various examples of these dualities. As an application, we bosonize fermionic s…
The Heisenberg evolution of a given unitary operator corresponds classically to a fixed canonical transformation that is viewed through a moving coordinate system. The operators that form the bases of the Weyl representation and its Fourier transform, the chord representation, are, respectively, unitary reflection and translation operators. Thus, the general semiclassical study of unitary operators allows us to propagate arbitrary operators, including density operators, i.e. the Wigner function. The various propagation kernels are different representations of the superoperators which act on th…
For the efficient implementation of quantum algorithms, practical ways to generate many-body entanglement are a basic requirement. Specifically, coupling multiple qubit pairs at once can be advantageous and can lead to multi-qubit operations useful in the construction of hardware-tailored algorithms. Here we harness the simultaneous coupling of qubits on a chain and engineer a set of non-local parity-dependent quantum operations suitable for a wide range of applications. The resulting effective long-range couplings directly implement a parametrizable Trotter-step for Jordan-Wigner fermions and…
Uncertainty and Wigner negativity in Hilbert-space classical mechanics
本文在希尔伯特空间经典力学中重现了量子力学的不确定性关系和威格纳负值特征。
Operational certification of nonclassicality in arbitrary quantum states from few copies
提出利用Wigner函数矩检测任意态Wigner负性的统一框架，通过少量副本估计矩，避免完整相空间层析，并扩展至纠缠检测。
Dunkl derivative enriches solutions by discussing parity due to its reflection operator. Very recently, one of the authors of this manuscript presented one of the most general forms of Dunkl derivative that depends on three Wigner parameters to have a better tuning. In this manuscript, we employ the latter generalized Dunkl derivative in a relativistic equation to examine two-dimensional harmonic and anharmonic oscillators solutions. We obtain the solutions by Nikiforov-Uvarov and QES methods, respectively. We show that degenerate states can occur according to the Wigner parameter values.
We investigate the capabilities of loss-tolerant quantum state characterization using a photon-number resolving, time-multiplexed detector (TMD). We employ the idea of probing the Wigner function point-by-point in phase space via photon parity measurements and displacement operations, replacing the conventional homodyne tomography. Our emphasis lies on reconstructing the Wigner function of non-Gaussian Fock states with highly negative values in a scheme that is based on a realistic experimental setup. In order to establish the concept of loss-tolerance for state characterization we show how lo…
Multiphoton heralding generates large-amplitude squeezed Schrödinger cat states and parity-selective Fock superpositions from squeezed vacuum via an OPA
提出利用光学参量放大器（OPA）的多光子触发方案，通过注入和检测光子实现高阶光子减法，从而生成大振幅压缩薛定谔猫态和低阶宇称选择性Fock叠加态，这些态具有强Wigner负性和高相空间复杂度，且在光子损失下仍保持复杂性，超越海森堡极限。
A possible two-qubit gate for optical quantum computing is the parity gate based on the weak Kerr effect. Two photonic qubits modulate the phase of a coherent state, and a quadrature measurement of the coherent state reveals the parity of the two qubits without destroying the photons. This can be used to create so-called cluster states, a universal resource for quantum computing. Here, the effect of self-phase modulation on the parity gate is studied, introducing generating functions for the Wigner function of a modulated coherent state. For materials with non-EIT-based Kerr nonlinearities, th…
Translation-invariant quantum low-density parity-check codes from compactified fracton models
本文研究了具有平移对称性和局部检查的量子纠错码，提出了一种统一框架，涵盖了多种平移不变码，包括许多分形码和所有阿贝尔双块群代数（A2BGA）码，包括双变量自行车（BB）码。
It has recently been shown that it is possible to represent the complete quantum state of any system as a phase-space quasi-probability distribution (Wigner function) [Phys Rev Lett 117, 180401]. Such functions take the form of expectation values of an observable that has a direct analogy to displaced parity operators. In this work we give a procedure for the measurement of the Wigner function that should be applicable to any quantum system. We have applied our procedure to IBM's Quantum Experience five-qubit quantum processor to demonstrate that we can measure and generate the Wigner function…
Reporting about the Wigner formalism for describing Dirac spinor structures through a covariant phase-space formulation, the quantum information quantifiers for purity and mutual information involving spin-parity (discrete) and position-momentum (continuous) degrees of freedom are consistently obtained. For Dirac spinor Wigner operators decomposed into Poincaré classes of $SU(2) \otimes SU(2)$ spinor couplings, a definitive expression for quantum purity is identified in a twofold way: firstly, in terms of phase-space positively defined quantities, and secondly, in terms of the {\em spin-parity…
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 ext…
Discriminating between quantum computing architectures that can provide quantum advantage from those that cannot is of crucial importance. From the fundamental point of view, establishing such a boundary is akin to pinpointing the resources for quantum advantage; from the technological point of view, it is essential for the design of non-trivial quantum computing architectures. Wigner negativity is known to be a necessary resource for computational advantage in several quantum-computing architectures, including those based on continuous variables (CVs). However, it is not a sufficient resource…
Operators in quantum mechanics - either observables, density or evolution operators, unitary or not - can be represented by c-numbers in operator bases. The position and momentum bases are in one to one correspondence with lagrangian planes in double phase space, but this is also true for the well known Wigner-Weyl correspondence based on translation and reflection operators. These phase space methods are here extended to the representation of superoperators. We show that the Choi-Jamiolkowsky isomorphism between the dynamical matrix and the linear action of the superoperator constitutes a "do…
Following the information-based approach to Dirac spinors under a constant magnetic field, the phase-space representation of symmetric and anti-symmetric localized Dirac cat states is obtained. The intrinsic entanglement profile implied by the Dirac Hamiltonian is then investigated so as to shed a light on quantum states as carriers of qubits correlated by phase-space variables. Corresponding to the superposition of Gaussian states, cat states exhibit non-trivial elementary information dynamics which include the interplay between intrinsic entanglement and quantum superposition as reported by …
Selective Filtering of Photonic Quantum Entanglement via Anti-Parity-Time Symmetry
本文提出基于反宇称-时间对称性的非厄米光子滤波方法，在无损耗波导网络中利用Lanczos变换实现任意输入态的高保真纠缠提取，并可扩展至多光子情形且对退相干具有鲁棒性。
Generation of Wigner functions of Landau levels and determination of their symmetries and generic properties are achieved in the autonomous framework of deformation quantization. Transformation properties of diagonal Wigner functions under space inversion, time reversal and parity transformations are specified and their invariance under a four-parameter subgroup of symplectic transformations are established. A generating function for all Wigner functions is developed and this has been identified as the phase-space coherent state for Landau levels. Integrated forms of generating function are us…
The parity of the number of elementary excitations present in a quantum system provides important insights into its physical properties. Parity measurements are used, for example, to tomographically reconstruct quantum states or to determine if a decay of an excitation has occurred, information which can be used for quantum error correction in computation or communication protocols. Here we demonstrate a versatile parity detector for propagating microwaves, which distinguishes between radiation fields containing an even or odd number n of photons, both in a single-shot measurement and without …
Experimental observation of parity-symmetry-protected phenomena in the quantum Rabi model with a trapped ion
该研究利用囚禁离子模拟可进入超强及深耦合区的扩展量子 Rabi 模型，实验揭示宇称对称性对声子激发、基态纠缠、量子叠加及相关可观测量的保护与调控作用。
The ``little group'' for massless particles (namely, the Lorentz transformations $Λ$ that leave a null vector invariant) is isomorphic to the Euclidean group E2: translations and rotations in a plane. We show how to obtain explicitly the rotation angle of E2 as a function of $Λ$ and we relate that angle to Berry's topological phase. Some particles admit both signs of helicity, and it is then possible to define a reduced density matrix for their polarization. However, that density matrix is physically meaningless, because it has no transformation law under the Lorentz group, even under ordinary…
Classical surfaces in phase space correspond to quantum states in Hilbert space. Subsystems specify factor spaces of the Hilbert space. An entangled state corresponds semiclassically to a surface that cannot be decomposed into a product of lower dimensional surfaces. Such a classical factorization never exists for ergodic eigenstates of a chaotic Hamiltonian. The space of quantum operators corresponds to a double phase space. The various representations of the density operator then result from alternative choices of allowed coordinate planes. In the case of the Wigner function and its Fourier …
Wigner representation enables the exact derivation of the atom interferometer phase, unlike the path integral approach
针对非惯性参考系中原子干涉仪相位推导问题，本文利用维格纳表示下的原子密度矩阵方程得到精确表达式（含多种力项），并在小旋转角和短探测时间极限下推导出归因于平台运动的三个新项。
Preparing squeezed, cat and GKP states with parity measurements
提出基于位移宇称测量的协议，通过少量测量即可高效制备压缩态、猫态和Gottesman-Kitaev-Preskill态，并展示其对实验缺陷的鲁棒性。
Phase Spaces, Parity Operators, and the Born-Jordan Distribution
本文提出利用（平移）宇称算符避免传统积分变换来刻画相空间分布函数，精确表征其数学存在性，并将其应用于Born-Jordan分布，通过确定广义谱分解实现了高效计算并揭示了其独特特征。
Parity-detection-based Mach-Zehnder interferometry with coherent and non-Gaussian squeezed vacuum states as inputs
本文理论上研究了基于宇称探测的Mach-Zehnder干涉仪在相位估计中引入非高斯操作的优势，发现光子扣除、添加和催化均能提升相位灵敏度，其中单光子添加为最佳操作。
Single and coupled quantum van der Pol oscillators: Wigner-function dynamics, exact qutrit solutions, and synchronization
该研究针对单耦合量子范德波尔振荡器，推导其维格纳函数演化方程，利用qutrit子空间简化模型分析同步特性，揭示了非线性耗散强度对同步区域的影响。
We present an operator approach to the description of photon polarization, based on Wigner's concept of elementary relativistic systems. The theory of unitary representations of the Poincare group, and of parity, are exploited to construct spinlike operators acting on the polarization states of a photon at each fixed energy momentum. The nontrivial topological features of these representations relevant for massless particles, and the departures from the treatment of massive finite spin representations, are highlighted and addressed.
Dissipative forces are ubiquitous and thus constitute an essential part of realistic physical theories. However, quantization of dissipation has remained an open challenge for nearly a century. We construct a quantum counterpart of classical friction, a velocity-dependent force acting against the direction of motion. In particular, a translationary invariant Lindblad equation is derived satisfying the appropriate dynamical relations for the coordinate and momentum (i.e., the Ehrenfest equations). Numerical simulations establish that the model approximately equilibrates. These findings signific…
Quantum pure noise-induced transitions: A truly nonclassical limit cycle sensitive to number parity
本文提出一个无经典极限的非线性阻尼量子振子模型，证明乘性噪声可诱导保守的非经典极限环，并具有Wigner负性和粒子数宇称敏感环流。
Orbital-Free Quasi-Density Functional Theory
本文提出无轨道准密度泛函理论，以准密度为变量计算单体维格纳准概率，规避泡利势与动能近似难题，并揭示其与约化密度矩阵泛函的深层联系及维格纳负性与v-可表示性的关系。
Highly tunable platforms for realizing topological phases of matter are emerging from atomic and photonic systems, and offer the prospect of designing interactions between particles. The shape of the potential, besides playing an important role in the competition between different fractional quantum Hall phases, can also trigger the transition to symmetry-broken phases, or even to phases where topological and symmetry-breaking order coexist. Here, we explore the phase diagram of an interacting bosonic model in the lowest Landau level at half-filling as two-body interactions are tuned. Apart fr…
Weighted Phase-Space Paths for Exact Wigner Dynamics
本文提出通过加权经验测度重构Wigner函数，以解决非二次势能下Wigner-Moyal生成器中高阶项的问题，并探讨其在非谐动力学中的应用。
Dissipative Majorana Chain---Anomalies and Non-Invertible Symmetries
该研究分析耗散马约拉纳链的对称性与反常，通过算子-态映射建立其Lindbladian的非厄米表述，结合格点玻色化与投影得到伊辛Lindbladian，为开放量子系统的相关扩展提供可控框架。
Quantum Tomography of Suspended Carbon Nanotubes
本文提出了一种全机械方法，用于悬浮碳纳米管的柔性模式的相干控制和量子态重建，通过原子力显微镜实现单点可控驱动，支持Rabi振荡和Ramsey干涉等双能级协议，并结合位移奇点采样实现Wigner函数成像，无需光学加热或专用微波驱动线。
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