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Quantum state exclusion for group-generated ensembles of pure states

Arnau Diebra, Santiago Llorens, Emili Bagan, Gael Sentís, Ramon Muñoz-Tapia

Abstract

Quantum state exclusion is the task of determining which states from a given set a system was not prepared in. We provide a complete solution to optimal quantum state exclusion for arbitrary sets of pure states generated by finite groups, establishing necessary and sufficient conditions for perfect (zero-error conclusive) exclusion. When perfect exclusion is impossible, we introduce two natural extensions: minimum-error and unambiguous exclusion. For both, we derive the optimal protocols and present analytical expressions for the corresponding failure probabilities and measurements, providing additional insight into how quantum states encode information.

Quantum state exclusion for group-generated ensembles of pure states

Abstract

Quantum state exclusion is the task of determining which states from a given set a system was not prepared in. We provide a complete solution to optimal quantum state exclusion for arbitrary sets of pure states generated by finite groups, establishing necessary and sufficient conditions for perfect (zero-error conclusive) exclusion. When perfect exclusion is impossible, we introduce two natural extensions: minimum-error and unambiguous exclusion. For both, we derive the optimal protocols and present analytical expressions for the corresponding failure probabilities and measurements, providing additional insight into how quantum states encode information.

Paper Structure

This paper contains 9 sections, 2 theorems, 88 equations, 1 figure.

Table of Contents

  1. Gram Matrix as a Complete Descriptor of Pure State Ensembles
  2. Representation Theory of Finite Groups: A Concise Summary
  3. Linear representations
  4. Projective representations
  5. Group-generated Gram matrices
  6. Linear representations
  7. Projective representations
  8. Optimality of minimum-error exclusion measurement
  9. Optimality of unambiguous exclusion POVM

Key Result

Lemma 1

The optimal POVM for QSE can always be chosen to be covariant with a rank-1 seed. Specifically, it takes the form $\{\Pi_g = U_g |\omega\rangle\langle \omega| U_g^\dagger \mid g \in \mathcal{G} \}$ for minimum-error QSE, with the additional POVM element $\Pi_Q=\hbox{\rm \openone}-\sum_{g\in{\cal G}} where the eigenstate $|v^1_1\rangle|u^1_1\rangle$ corresponds to the largest eigenvalue of $\Omega$

Figures (1)

  • Figure 1: Regular tetrahedron generated by $\mathfrak{G} = \{\openone, \mathcal{R}_x(\pi), \mathcal{R}_y(\pi), \mathcal{R}_z(\pi)\}$ acting on $(1,1,1)$.

Theorems & Definitions (3)

  • Lemma 1
  • proof
  • Theorem I