A classical interpretation of the Scrooge distribution

TL;DR

This paper provides a classical interpretation of the quantum-inspired Scrooge distribution, especially for real amplitudes, linking quantum concepts to classical probability over the simplex.

Contribution

It introduces a natural classical interpretation for the real-amplitude Scrooge distribution and explores the transition to the complex-amplitude case, illuminating quantum-classical relations.

Findings

Classical interpretation for real-amplitude Scrooge distribution established

Transition to complex amplitudes involves a non-trivial step

Insights into quantum and classical probability connection

Abstract

The Scrooge distribution is a probability distribution over the set of pure states of a quantum system. Specifically, it is the distribution that, upon measurement, gives up the least information about the identity of the pure state, compared with all other distributions having the same density matrix. The Scrooge distribution has normally been regarded as a purely quantum mechanical concept, with no natural classical interpretation. In this paper we offer a classical interpretation of the Scrooge distribution viewed as a probability distribution over the probability simplex. We begin by considering a real-amplitude version of the Scrooge distribution, for which we find that there is a non-trivial but natural classical interpretation. The transition to the complex-amplitude case requires a step that is not particularly natural but that may shed light on the relation between quantum mechanics and classical probability theory.