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Rényi Entropy, Signed Probabilities, and the Qubit

Adam Brandenburger, Pierfrancesco La Mura, Stuart Zoble

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 Rényi entropy (a generalization of Shannon entropy) suitably defined for the signed phase-space probability distributions that arise in representing quantum states.

Rényi Entropy, Signed Probabilities, and the Qubit

Abstract

The states of the qubit, the basic unit of quantum information, are 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 Rényi entropy (a generalization of Shannon entropy) suitably defined for the signed phase-space probability distributions that arise in representing quantum states.

Paper Structure

This paper contains 5 sections, 1 theorem, 63 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Rényi Entropy
  4. Main Theorem
  5. Conclusion

Key Result

Theorem 1

The potential quantum states satisfying the Uncertainty Principle are precisely the states of the qubit.

Theorems & Definitions (4)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1