We study the Weyl-Wigner transform in the case of discrete variables defined in a Hilbert space of finite prime-number dimensionality $N$. We define a family of Weyl-Wigner transforms as function of a phase parameter. We show that it is only for a sp
ecific value of the parameter that all the properties we have examined have a parallel with the case of continuous variables defined in an infinite-dimensional Hilbert space. A geometrical interpretation is briefly discussed.
Measurements on classical systems are usually idealized and assumed to have infinite precision. In practice, however, any measurement has a finite resolution. We investigate the theory of non-ideal measurements in classical mechanics using a measurem
ent probe with finite resolution. We use the von Neumann interaction model to represent the interaction between system and probe. We find that in reality classical systems are affected by measurement in a similar manner as quantum systems. In particular, we derive classical equivalents of Luders rule, the collapse postulate, and the Lindblad equation.
The experimental realization of successive non-demolition measurements on single microscopic systems brings up the question of ergodicity in Quantum Mechanics (QM). We investigate whether time averages over one realization of a single system are rela
ted to QM averages over an ensemble of similarly prepared systems. We adopt a generalization of von Neumann model of measurement, coupling the system to $N$ probes --with a strength that is at our disposal-- and detecting the latter. The model parallels the procedure followed in experiments on Quantum Electrodynamic cavities. The modification of the probability of the observable eigenvalues due to the coupling to the probes can be computed analytically and the results compare qualitatively well with those obtained numerically by the experimental groups. We find that the problem is not ergodic, except in the case of an eigenstate of the observable being studied.
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 Luders 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 Wigners formula emerges in the strong-coupling limit, while Kirkwoods 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.
Wigner function is a quasi-distribution 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 Wigner function and appropriate measurements invo
lving the system 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 position and momentum. We show that one can relate Wigner function to Kirkwood joint quasi-distribution 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.
