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Asymptotic Quantum State Discrimination for Mixtures of Unitarily Related States

Alberto Acevedo, Janek Wehr

Abstract

Given a mixture of states, finding a way to optimally discriminate its elements is a prominent problem in quantum communication theory. In this paper, we will address mixtures of density operators that are unitarily equivalent via elements of a one-parameter unitary group, and the corresponding quantum state discrimination (QSD) problems. We will be particularly interested in QSD as time goes to infinity. We first present an approach to QSD in the case of countable mixtures and address the respective asymptotic QSD optimization problems, proving necessary and sufficient conditions for minimal error to be obtained in the asymptotic regime (we say that in such a case QSD is fully solvable). We then outline an analogous approach to uncountable mixtures, presenting some conjectures that mirror the results presented for the cases of countable mixtures. As a technical tool, we prove and use an infinite dimensional version of the well-known Barnum-Knill bound.

Asymptotic Quantum State Discrimination for Mixtures of Unitarily Related States

Abstract

Given a mixture of states, finding a way to optimally discriminate its elements is a prominent problem in quantum communication theory. In this paper, we will address mixtures of density operators that are unitarily equivalent via elements of a one-parameter unitary group, and the corresponding quantum state discrimination (QSD) problems. We will be particularly interested in QSD as time goes to infinity. We first present an approach to QSD in the case of countable mixtures and address the respective asymptotic QSD optimization problems, proving necessary and sufficient conditions for minimal error to be obtained in the asymptotic regime (we say that in such a case QSD is fully solvable). We then outline an analogous approach to uncountable mixtures, presenting some conjectures that mirror the results presented for the cases of countable mixtures. As a technical tool, we prove and use an infinite dimensional version of the well-known Barnum-Knill bound.
Paper Structure (7 sections, 16 theorems, 107 equations)

This paper contains 7 sections, 16 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Some Important Theorems
  3. Countable Mixtures of Unitarily Related Families.
  4. Spectral Decomposition and Spectral Measures.
  5. Unitarily Related Linear Combinations of Finite Mixtures.
  6. Uncountable Mixtures.
  7. Acknowledgements

Key Result

Theorem 1

Hellström Bound: Let $\mathscr{H}$ be an arbitrary $Hilbert$ space. For any mixture of the form we have Here $\|\mathbf{\hat{A}}\|_{1}:=Tr\{\sqrt{\mathbf{\hat{A}}^{\dagger}\mathbf{\hat{A}}}\}$ (Trace norm).

Theorems & Definitions (32)

  • Definition 1
  • Remark 1
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Lemma 1
  • proof
  • ...and 22 more