R\'enyi Entropy, Signed Probabilities, and the Qubit

TL;DR

This paper explores the characterization of qubit states using an entropic uncertainty principle based on Rényi entropy within a signed phase space framework, advancing the axiomatization of quantum mechanics.

Contribution

It introduces a novel approach to describe qubit states through Rényi entropy and signed probabilities, contributing to the axiomatization of quantum mechanics.

Findings

Characterization of qubit states via entropic uncertainty principles.

Use of Rényi entropy for signed phase-space distributions.

Advancement in axiomatizing quantum mechanics.

Abstract

The states of the qubit, the basic unit of quantum information, are $2 \times 2$ positive semi-definite Hermitian matrices with trace 1. We contribute to the program to axiomatize quantum mechanics by characterizing these states in terms of an entropic uncertainty principle formulated on an eight-point phase space. We do this by employing R\'enyi entropy (a generalization of Shannon entropy) suitably defined for the signed phase-space probability distributions that arise in representing quantum states.