Discrete Wigner Function Research Articles

Quantum spin tunneling in a soluble quasi-spin model: an angle-based potential description

We propose an approach which allows one to construct and use a potential function written in terms of an angle variable to describe interacting spin systems. We show how this can be implemented in the Lipkin–Meshkov–Glick, here considered a paradigmatic spin model. It is shown how some features of the energy gap can be interpreted in terms of a spin tunneling. A discrete Wigner function is constructed for a symmetric combination of two states of the model and its time evolution is obtained. The physical information extracted from that function reinforces our description of phase oscillations in a potential.

Discrete phase space based on finite fields

The original Wigner function provides a way of representing in phase space the quantum states of systems with continuous degrees of freedom. Wigner functions have also been developed for discrete quantum systems, one popular version being defined on a 2N x 2N discrete phase space for a system with N orthogonal states. Here we investigate an alternative class of discrete Wigner functions, in which the field of real numbers that labels the axes of continuous phase space is replaced by a finite field having N elements. There exists such a field if and only if N is a power of a prime; so our formulation can be applied directly only to systems for which the state-space dimension takes such a value. Though this condition may seem limiting, we note that any quantum computer based on qubits meets the condition and can thus be accommodated within our scheme. The geometry of our N x N phase space also leads naturally to a method of constructing a complete set of N+1 mutually unbiased bases for the state space.

Quantum phase space points for Wigner functions in finite-dimensional spaces

We introduce quantum states associated with single phase space points in the Wigner formalism for finite-dimensional spaces. We consider both continuous and discrete Wigner functions. This analysis provides a procedure for a direct practical observation of the Wigner functions for states and transformations without inversion formulas.

Nonclassical statistical properties of finite-coherent states in the framework of the Jaynes–Cummings model

Adopting the framework of the nonresonant Jaynes–Cummings model, we investigate the nonclassical statistical properties of coherent states defined in a finite-dimensional Hilbert space, considering the single-mode cavity field prepared in a finite and discrete harmonic oscillator-like coherent state with a small average number of photons. Explicit expressions for the time evolution of various functions characterizing the quantum state, such as the Mandel's Q parameter, the photon-number distribution and its respective entropy, the discrete Wigner function, and the number-phase uncertainty relation, are investigated in detail. We also show that the atomic inversion possesses regular structures with collapses and revivals in the Rabi oscillations since the detuning between atom and field is large as compared to the coupling constant, i.e., ( κ/2 g) 2⪢1. The numerical and analytical results obtained in this work turn evident the quantum interference effects between the components of the finite-coherent states. Furthermore, we present a discussion about unitary depolarizers in finite-dimensional Hilbert space and their connection to quantum information theory.

Discrete Wigner functions and the phase-space representation of quantum teleportation

We present a phase-space description of the process of quantum teleportation for a system with an N-dimensional space of states. For this purpose we define a discrete Wigner function which is a minor variation of previously existing ones. This function is useful to represent a composite quantum system in phase space and to analyze situations where entanglement between subsystems is relevant (dimensionality of the space of states of each subsystem is arbitrary). We also describe how a direct tomographic measurement of this Wigner function can be performed.

Quantum computers in phase space

We represent both the states and the evolution of a quantum computer in phase space using the discrete Wigner function. We study properties of the phase space representation of quantum algorithms: apart from analyzing important examples, such as the Fourier Transform and Grover's search, we examine the conditions for the existence of a direct correspondence between quantum and classical evolutions in phase space. Finally, we describe how to directly measure the Wigner function in a given phase space point by means of a tomographic method that, itself, can be interpreted as a simple quantum algorithm.

Reconstructing the discrete Wigner function and some properties of the measurement bases

We derive a direct reconstruction algorithm for the discrete Wigner function through different types of measurements. For a system described in a Hilbert space of dimension ${N=N}_{1}\dots{}{N}_{p},$ where the numbers ${N}_{i}$ are prime, the reconstruction is accomplished with ${(N}_{1}+1)\dots{}{(N}_{p}+1)$ factorable (local) von Neumann measurements. For the special case where the dimension is a power of a prime, the reconstruction can be performed in a much more efficient way using $N+1$ complementary measurements. If the system is composed of a number of smaller subsystems, these measurements will then in general be nonseparable.

An expectation value expansion of Hermitian operators in a discrete Hilbert space

We discuss a real-valued expansion of any Hermitian operator defined in a Hilbert space of finite dimension N, where N is a prime number, or an integer power of a prime. The expansion has a direct interpretation in terms of the operator expectation values for a set of complementary bases. The expansion can be said to be the complement of the discrete Wigner function. We expect the expansion to be of use in quantum information applications since qubits typically are represented by a discrete, and finite-dimensional physical system of dimension N=2^p, where p is the number of qubits involved. As a particular example we use the expansion to prove that an intermediate measurement basis (a Breidbart basis) cannot be found if the Hilbert space dimension is 3 or 4.

The canonical Kravchuk basis for discrete quantum mechanics

The well known Kravchuk formalism of the harmonic oscillator obtained from the direct discretization method is shown to be a new way of formulating discrete quantum phase space. It is shown that the Kravchuk oscillator Hamiltonian has a well defined unitary canonical partner which we identify with the quantum phase of the Kravchuk oscillator. The generalized discrete Wigner function formalism based on the action and angle variables is applied to the Kravchuk oscillator and its continuous limit is examined.

Quantum-State Tomography and Discrete Wigner Function

Quantum-State Tomography and Discrete Wigner Function

Discrete Wigner function and quantum-state tomography.

The theory of discrete Wigner functions and of discrete quantum-state tomography [U. Leonhardt, Phys. Rev. Lett. 74, 4101 (1995)] is studied in more detail guided by the picture of precession tomography. Odd- and even-dimensional systems (angular momenta and spins, bosons, and fermions) are considered separately. Relations between simple number theory and the quantum mechanics of finite-dimensional systems are pointed out. In particular, the multicomplementarity of the precession states distinguishes prime dimensions from composite ones. \textcopyright{} 1996 The American Physical Society.

Quantum-state tomography and discrete Wigner function.

A tomographical scheme is proposed to infer the quantum states of finite--dimensional systems from experiments. For this a new discrete Wigner formalism is developed.

Symmetries and time evolution in discrete phase spaces: a soluble model calculation

Using the flexibility and constructive definition of the Schwinger bases, we developed different mapping procedures to enhance different aspects of the dynamics and of the symmetries of an extended version of the two-level Lipkin model. The classical limits of the dynamics are discussed in connection with the different mappings. Discrete Wigner functions are also calculated.

Fokker-Planck equation associated with the Wigner function of a quantum system with a finite number of states

The authors consider a quantum system with a finite number N of states and show that a Markov process evolving in an 'extended' discrete phase space can be associated with the discrete Wigner function of the system. This Wigner function is built using the Weyl quantisation procedure on the group ZN*ZN. Moreover, the authors can use this process to compute the quantum mean values as probabilistic expectations of functions of this process. This probabilistic formulation can be seen as a stochastic mechanics in phase space.