Transition probabilities and transition rates in discrete phase space

TL;DR

This paper explores the properties of transition probabilities and rates in discrete quantum phase space, providing criteria for their legitimacy and methods to compute them for Hamiltonian evolutions.

Contribution

It introduces simple criteria to determine when transition probabilities and rates correspond to valid quantum processes and offers a method to compute transition rates from Hamiltonians in discrete phase space.

Findings

Derived criteria for legitimate quantum transition probabilities.

Provided a method to compute transition rates from Hamiltonians.

Analyzed the formal similarity between quantum evolution and probabilistic processes.

Abstract

The evolution of the discrete Wigner function is formally similar to a probabilistic process, but the transition probabilities, like the discrete Wigner function itself, can be negative. We investigate these transition probabilities, as well as the transition rates for a continuous process, aiming particularly to give simple criteria for deciding when a set of such quantities corresponds to a legitimate quantum process. We also show how the transition rates for any Hamiltonian evolution can be worked out by expanding the Hamiltonian as a linear combination of displacement operators in the discrete phase space.