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arXiv:2007.10187 · Physical Review A

Transition probabilities and transition rates in discrete phase space

William F. Braasch, William K. Wootters

Published Nov 4, 2020• 4 citations
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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.

DOI
10.1103/physreva.102.052204

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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.
The main aspects of a discrete phase space formalism are presented and the discrete dynamical bracket, suitable for the description of time evolution in finite-dimensional spaces, is discussed. A set of operator bases is defined in such a way that the Weyl-Wigner formalism is shown to be obtained as a limiting case. In the same form, the Moyal bracket is shown to be the limiting case of the discrete dynamical bracket. The dynamics in quantum discrete phase spaces is shown not to be attained from discretization of the continuous case.
In this paper we give a partial answer to the problem: When does an initially non-negative Wigner function remain non-negative under the effect of the time evolution? We show that, for pure states, this is the case for linear systems only; to prove this we use the fact that the Wigner function is non-negative if and only if the wavefunction is Gaussian. We also prove that the Green’s solution of the evolution equation of the Wigner function, which in the framework of probability theory corresponds to the conditional probability density, takes on negative values. We utilize a theorem, about moments, borrowed from Pawula. We conclude that the Wigner phase-space formulation of quantum mechanics cannot receive a genuine probabilistic interpretation.
We study the class of discrete Wigner functions proposed by Gibbons et al. [Phys. Rev. A 70, 062101 (2004)] to describe quantum states using a discrete phase space based on finite fields. We find the extrema of such functions for small Hilbert-space dimensions and present a quantum-information application: a construction of quantum random-access codes. These are constructed using the complete set of phase-space point operators to find encoding states and to obtain the codes' average success rates for Hilbert-space dimensions 2, 3, 4, 5, 7, and 8.
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 measure directly 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.
A classical simulation scheme of quantum computation given a restricted set of states and measurements may be---occasionally, but only occasionally---interpreted naturally as a statistical simulation of positive quasi-probability distributions on a phase space. In this dissertation, we explore phase space representations for finite-dimensional quantum systems and their negativities beyond the usual analogues of the Wigner function. The first line of study focuses on a characterization tool for valid quasi-probability distributions of (possibly mixed) quantum states. A quantum generalization of Bochner's theorem from classical probability theory simultaneously characterizes both the set of valid Wigner functions and the subset of positive ones. We extend this theorem to discrete phase spaces based on projective representations of abelian groups, including the discrete Wigner functions of Gross (J. Math. Phys. 47, 122107 (2006)) and Gibbons et al. (Phys. Rev. A 70, 062101 (2004)). The rest of the dissertation is then dedicated to phase space representations covariant with respect to a general Lie group. This means, in particular, that operators that lie in the same orbit of the group have the same amount of negativity in their quasi-probabilities---a natural requirement if one would like to interpret negativity as non-classicality, and the group action as a classical dynamics. We show that the construction due to Brif and Mann (Phys. Rev. A 59, 971 (1999)) of group covariant phase space representations satisfying the so-called Stratonovich-Weyl correspondence is essentially unique, given the group and an input state in the relevant representation. As an application, we construct quasi-probability representations on a compact phase space of fermionic Gaussian states tailored to a classically simulatable problem of fermionic linear optics and find that, in the first non-trivial case of four fermionic modes, the quasi-probabilities of the Gaussian states exhibit a considerable amount of negativity.