A Wigner-function formulation of finite-state quantum mechanics

For a non-relativistic system with only continous degrees of freedom (no spin, for example), the original Wigner function can be used as an alternative to the density matrix to represent an arbitrary quantum state. Indeed, the quantum mechanics of such systems can be formulated entirely in terms of the Wigner function and other functions on phase space, with no mention of state vectors or operators. In the present paper this Wigner-function formulation is extended to systems having only a finite number of orthogonal states. The “phase space” for such a system is taken to be not continuous but discrete. In the simplest cases it can be pictured as an N×N array of points, where N is the number of orthogonal states. The Wigner function is a real function on this phase space, defined so that its properties are closely analogous to those of the original Wigner function. In this formulation, observables, like states, are represented by real functions on the discrete phase space. The complex numbers still play an important role: they appear in an essential way in the rule for forming composite systems.     

Tomographic characteristics of spin states

Spin states are studied in the tomographic-probability representation. The standard probability distribution of spin projection onto a direction in space is used instead of the spinor or the density matrix to identify the quantum state. The Shannon entropy and information are associated with the spin tomographic probability. A short review of the probability-theory notions is presented. Analysis of tomographic entropy and tomographic information for the Werner state is considered. The probability representation is used to describe a spin-3/2 particle and two qubits. The connection of tomographic entropy with the von Neumann entropy is discussed.     

Tomographic probability representation for quantum fermion fields

Tomographic probability representation is introduced for fermion fields. The states of the fermions are mapped onto the probability distribution of discrete random variables (spin projections). The operators acting on the fermion states are described by fermionic tomographic symbols. The product of the operators acting on the fermion states is mapped onto the star-product of the fermionic symbols. The kernel of the star-product is obtained. The antisymmetry of the fermion states is formulated as a specific symmetry property of the tomographic joint probability distribution associated with the states.     

Wigner function for discrete phase space: Exorcising ghost images

We construct, using simple geometrical arguments, a Wigner function defined on a discrete phase space of arbitrary integer Hilbert-space dimension that is free of redundancies. “Ghost images” plaguing other Wigner functions for discrete phase spaces are thus revealed as artifacts. It allows to devise a corresponding phase-space propagator in an unambiguous manner. We apply our definitions to eigenstates and propagator of the quantum baker map. Scars on unstable periodic points of the corresponding classical map become visible with unprecedented resolution.     

自旋1/2非对易朗道问题的Wigner函数(英文)

Wigner函数在对量子体系状态的描述方面具有重要的意义。 讨论了自旋1/2非对易朗道问题的Wigner函数。首先回顾了对易空间中Wigner函数所服从的星本征方程, 然后给出了非对易相空间中自旋1/2朗道问题的Hamiltonian, 最后利用星本征方程（Moyal 方程）计算了非对易相空间中自旋1/2朗道问题具有矩阵表示形式的Wigner函数及其能级。With great significance in describing the state of quantum system, the Wigner function of the spin half non commutative Landau problem is studied in this paper. On the basis of the review of the Wigner function in the commutative space, which is subject to the *eigenvalue equation, Hamiltonian of the spin half Landau problem in the non commutative phase space is given. Then, energy levels and Wigner functions in the form of a matrix of the spin half Landau problem in the non commutative phase space are obtained by means of the _*-eigenvalue equation (or Moyal equation).     

Quantum tomography with wavelet transform in Banach space on homogeneous space

In this study the intimate connection is established between the Banach space wavelet reconstruction method on homogeneous spaces with both singular and nonsingular vacuum vectors, and some of the well known quantum tomographies, such as: Moyal-representation for a spin, discrete phase space tomography, tomography of a free particle, Homodyne tomography, phase space tomography and SU(1,1) tomography. And both the atomic decomposition and the Banach frame nature of these quantum tomographic examples are also revealed in details. Finally the connection between the wavelet formalism on Banach space and Q-function is discussed.     

Schwinger Bose实现下自旋相干态Wigner函数的特性分析

在Schwinger Bose实现下,引入纠缠态表象及Wigner算符在该表象下的表示,得到自旋相干态的Wigner函数,数值计算画出相空间中Wigner函数的分布图,并加以分析,发现在SchwingerBose实现下自旋相干态确实体现出纠缠特性。     

Discrete and generalized phase space techniques in critical quantum spin chains

We apply the Wigner function formalism from quantum optics via two approaches, Wootters' discrete Wigner function and the generalized Wigner function, to detect quantum phase transitions in critical spin-$\frac{1}{2}$ systems. We develop a general formula relating the phase space techniques and the thermodynamical quantities of spin models, which we apply to single, bipartite and multi-partite systems governed by the XY and the XXZ models. Our approach allows us to introduce a novel way to represent, detect, and distinguish first-, second- and infinite-order quantum phase transitions. Furthermore, we show that the factorization phenomenon of the XY model is only directly detectable by quantities based on the square root of the bipartite reduced density matrix. We establish that phase space techniques provide a simple, experimentally promising tool in the study of many-body systems and we discuss their relation with measures of quantum correlations and quantum coherence.     

Radon transforms of the Wigner operator on hyperplanes

The generalization of tomographic maps to hyperplanes is considered. We find that the Radon transform of the Wigner operator in multi-dimensional phase space leads to a normally ordered operator in binomial distribution---a mixed-state density operator. Reconstruction of the Wigner operator is also feasible. The normally ordered form and the Weyl ordered form of the Wigner operator are used in our derivation. The operator quantum tomography theory is expressed in terms of some operator identities, with the merit of revealing the essence of the theory in a simple and concise way.     

Quantum Phase Space from Schwinger’s Measurement Algebra

Schwinger’s algebra of microscopic measurement, with the associated complex field of transformation functions, is shown to provide the foundation for a discrete quantum phase space of known type, equipped with a Wigner function and a star product. Discrete position and momentum variables label points in the phase space, each taking \(N\) distinct values, where \(N\) is any chosen prime number. Because of the direct physical interpretation of the measurement symbols, the phase space structure is thereby related to definite experimental configurations.     

Chebyshev polynomials and Fourier transform of <Emphasis Type="Italic">SU</Emphasis>(2) irreducible representation character as spin tomographic star-product kernel

Spin tomographic symbols of qudit states and spin observables are studied. Spin observables are associated with the functions on a manifold whose points are labeled by the spin projections and sphere S ² coordinates. The star-product kernel for such functions is obtained in an explicit form and connected with the Fourier transform of characters of the SU(2) irreducible representation. The kernels are shown to be in close relation to the Chebyshev polynomials. Using specific properties of these polynomials, we establish the recurrence relation between the kernels for different spins. Employing the explicit form of the star-product kernel, a sum rule for Clebsch–Gordan and Racah coefficients is derived. Explicit formulas are obtained for the dual tomographic star-product kernel as well as for intertwining kernels which relate spin tomographic symbols and dual tomographic symbols.     

Probability Representation of Quantum Observables and Quantum States

We introduce the probability distributions describing quantum observables in conventional quantum mechanics and clarify their relations to the tomographic probability distributions describing quantum states. We derive the evolution equation for quantum observables (Heisenberg equation) in the probability representation and give examples of the spin-1/2 (qubit) states and the spin observables. We present quantum channels for qubits in the probability representation.     

Quantum chaos in the Wigner representation

The dynamics of a familiar model of stochastic behavior-the quantum kicked rotator-is analyzed in the Wigner representation. Exact nonlocal maps defined on a discrete phase space are derived. The basic dynamics of a quantum kicked rotator can be described satisfactorily by means of a simplified map that incorporates only the discrete nature of the phase space.     

The Phase Space Model of Nonrelativistic Quantum Mechanics

We focus on several questions arising during the modelling of quantum systems on a phase space. First, we discuss the choice of phase space and its structure. We include an interesting case of discrete phase space. Then, we introduce the respective algebras of functions containing quantum observables. We also consider the possibility of performing strict calculations and indicate cases where only formal considerations can be performed. We analyse alternative realisations of strict and formal calculi, which are determined by different kernels. Finally, two classes of Wigner functions as representations of states are investigated.     

Tomographic and Improved Subadditivity Conditions for Two Qubits and a Qudit with j = 3/2

We obtain a new entropic inequality for quantum and tomographic Shannon information for systems of two qubits. We derive the inequality relating quantum information and spin-tomographic information for particles with spin j = 3/2. We recommend the method for obtaining new entropic and information inequalities for composite systems of qudits, as well as for one qudit.     

Nonnegative Discrete Symbols and Their Probabilistic Interpretation

We review the quantizer–dequantizer formalism of constructing symbols of the density operators and quantum observables, such as Wigner functions and tomographic-probability distributions. We present a tutorial consideration of the technique of obtaining minimal sets of dequantizers (quorum) related to the observable eigenvalues for one-qubit states. We discuss a generalization of the quantizer–dequantizer scheme on the example of spin-1/2 states. We consider the possibilities of extending the results to two-qubit systems using spin tomograms of the state density matrix.     

The Negativity‐to‐Violation Map between Wigner Function and Quantum Contextuality Inequality for a Single Qudit

The relation between discrete Wigner function and quantum contextuality based on graph theory has been studied, following the work in [Nature 510,351(2014)]. To do this, non‐stabilizer projectors have been introduced to a series of non‐contextuality graphs based on stabilizer projectors for a single qudit with odd prime dimension. It has been found that, for a phase space point defined by Wootters, there exists a given set of states for an odd‐prime qudit where the negative discrete Wigner function on the phase space point means its quantum contextuality under measurements on the graphs designed by a specific method. To implement this method, a subset of non‐stabilizer projectors has been found. In the union of the set of states for all phase space points, there exists a negativity‐to‐violation map between Wigner function and quantum contextuality inequality. The robustness of the equivalence under depolarizing noise has been analyzed and discussed. For demonstration purposes, the graphs with different independence numbers and the corresponding set of states have been established on a single qutrit. Different to the cited work, this method involves only a single qudit, then is experimentally feasible for a qutrit.