Propensities in discrete phase spaces:<i>Q</i>function of a state in a finite-dimensional Hilbert space

Abstract

We present a Q function of a state of a quantum-mechanical system in a finite-dimensional Hilbert space. This discrete Q function is defined with the help of the W\'odkiewicz concept of propensities, i.e., we define the Q function as a discrete convolution of two Wigner functions based on Wootter's formalism, one of the state itself and one of the filter state. The discrete Q function takes nonnegative values in all ``points'' of the discrete phase space and is normalized and it is possible to reconstruct from it the density operator of the state under consideration. We analyze Q-function graphs for several states of interest.