Wigner function is a ``quasidistribution'' that provides a representation of the state of a quantum mechanical system in the phase space of position and momentum. In this paper we find a relation between the Wigner function and appropriate measurements involving the system's position and momentum which generalize the von Neumann model of measurement. We introduce two probes coupled successively in time to projectors associated with the system's position and momentum. We show that one can relate the Wigner function to the Kirkwood joint quasidistribution of position and momentum, the latter, in turn, being a particular case of successive measurements. We first consider the case of a quantum mechanical system described in a continuous Hilbert space and then turn to the case of a discrete, finite-dimensional Hilbert space.
Wigner function is a ``quasidistribution'' that provides a representation of the state of a quantum mechanical system in the phase space of position and momentum. In this paper we find a relation between the Wigner function and appropriate measurements involving the system's position and momentum which generalize the von Neumann model of measurement. We introduce two probes coupled successively in time to projectors associated with the system's position and momentum. We show that one can relate the Wigner function to the Kirkwood joint quasidistribution of position and momentum, the latter, in turn, being a particular case of successive measurements. We first consider the case of a quantum mechanical system described in a continuous Hilbert space and then turn to the case of a discrete, finite-dimensional Hilbert space.
We describe how to obtain information on a quantum-mechanical system by coupling it to a probe and detecting some property of the latter, using a model introduced by von Neumann, which describes the interaction of the system proper with the probe in a dynamical way. We first discuss single measurements, where the system proper is coupled to one probe with arbitrary coupling strength. The goal is to obtain information on the system detecting the probe position. We find the reduced density operator of the system, and show how Lüders rule emerges as the limiting case of strong coupling. The von Neumann model is then generalized to two probes that interact successively with the system proper. Now we find information on the system by detecting the position-position and momentum-position correlations of the two probes. The so-called "Wigner's formula" emerges in the strong-coupling limit, while "Kirkwood's quasi-probability distribution" is found as the weak-coupling limit of the above formalism. We show that successive measurements can be used to develop a state-reconstruction scheme. Finally, we find a generalized transform of the state and the observables based on the notion of successive measurements.
We study the Wigner Transform (WT) in the case of an infinite-dimensional continuous Hilbert space and a discrete Hilbert space of finite prime-number dimensionality N. In the discrete case we define a family of WT's as a function of a phase parameter. It is only for a specific value of the parameter that all the properties we have examined, including the matrix elements of the phase-space operator P and the relation between Wigner's function and Kirkwood's distribution, have a parallel in both types of Hilbert space. This close parallel was found constructing the two WT's in terms of a complete set of operators having similar properties, in the sense that the position and momentum operators appear disentangled in the two situations. We give a geometric interpretation of the properties involving the phase-space operator in the discrete case.
We study the quantum-mechanical uncertainty relation originating from the successive measurement of two observables A^ and B^, with eigenvalues an and bm, respectively, performed on the same system. We use an extension of the von Neumann model of measurement, in which two probes interact with the same system proper at two successive times, so we can exhibit how the disturbing effect of the first interaction affects the second measurement. Detecting the statistical properties of the second {\em probe} variable Q2 conditioned on the first {\em probe} measurement yielding Q1 we obtain information on the statistical distribution of the {\em system} variable bm conditioned on having found the system variable an in the interval δa around a(n). The width of this statistical distribution as function of δa constitutes an {\em uncertainty relation}. We find a general connection of this uncertainty relation with the commutator of the two observables that have been measured successively. We illustrate this relation for the successive measurement of position and momentum in the discrete and in the continuous cases and, within a model, for the successive measurement of a more general class of observables.
We define a Wigner distribution function for a one-dimensional finite quantum system, in which the position and momentum operators have a finite (multiplicity-free) spectrum. The distribution function is thus defined on discrete phase-space, i.e. on a finite discrete square grid. These discrete Wigner functions possess a number of properties similar to the Wigner function for a continuous quantum system such as the quantum harmonic oscillator. As an example, we consider the so-called oscillator model in dimension 2 j + 1, which is known to tend to the canonical oscillator when j tends to infinity. In particular, we compare plots of our discrete Wigner functions for the oscillator with the well known plots of Wigner functions for the canonical quantum oscillator. This comparison supports our approach to discrete Wigner functions.
Abstract There are several mathematical formulations of quantum mechanics. The Schrödinger picture expresses quantum states in terms of wavefunctions over, e.g. position or momentum. Alternatively, phase-space formulations represent states with quasi-probability distributions over, e.g. position and momentum. A quasi-probability distribution resembles a probability distribution but may have negative and non-real entries. The most famous quasi-probability distribution, the Wigner function, has played a pivotal role in the development of a continuous-variable quantum theory that has clear analogues of position and momentum. However, the Wigner function is ill-suited for much modern quantum-information research, which is focused on finite-dimensional systems and general observables. Instead, recent years have seen the Kirkwood–Dirac (KD) distribution come to the forefront as a powerful quasi-probability distribution for analysing quantum mechanics. The KD distribution allows tools from statistics and probability theory to be applied to problems in quantum-information processing. A notable difference to the Wigner function is that the KD distribution can represent a quantum state in terms of arbitrary observables. This paper reviews the KD distribution, in three parts. First, we present definitions and basic properties of the KD distribution and its generalisations. Second, we summarise the KD distribution’s extensive usage in the study or development of measurement disturbance; quantum metrology; weak values; direct measurements of quantum states; quantum thermodynamics; quantum scrambling and out-of-time-ordered correlators; and the foundations of quantum mechanics, including Leggett–Garg inequalities, the consistent-histories interpretation and contextuality. We emphasise connections between operational quantum advantages and negative or non-real KD quasi-probabilities. Third, we delve into the KD distribution’s mathematical structure. We summarise the current knowledge regarding the geometry of KD-positive states (the states for which the KD distribution is a classical probability distribution), describe how to witness and quantify KD non-positivity, and outline relationships between KD non-positivity, coherence and observables’ incompatibility.
We define a Wigner distribution function for a one-dimensional finite quantum system, in which the position and momentum operators have a finite (multiplicity-free) spectrum. The distribution function is thus defined on discrete phase-space, i.e. on a finite discrete square grid. These discrete Wigner functions possess a number of properties similar to the Wigner function for a continuous quantum system such as the quantum harmonic oscillator. As an example, we consider the so-called su(2) oscillator model in dimension 2j + 1, which is known to tend to the canonical oscillator when j tends to infinity. In particular, we compare plots of our discrete Wigner functions for the su(2) oscillator with the well known plots of Wigner functions for the canonical quantum oscillator. This comparison supports our approach to discrete Wigner functions. 1