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Conclusive exclusion of quantum states with group action

Hongshun Yao, Xin Wang

TL;DR

This work analyzes conclusive single-state exclusion under group actions, linking symmetry to fundamental limits on quantum information extraction. By leveraging the isotypical decomposition of the representation and seed-state amplitudes, it derives a general sufficient condition for conclusive exclusion applicable to finite and compact Lie groups, and a tight necessary-and-sufficient condition for Abelian groups that recovers a generalized PBR result. A constructive POVM is provided, enabling perfect exclusion when the amplitude inequality $d_{\mu_0}|a_{\mu_0}|\le\sum_{\mu\neq\mu_0} d_\mu|a_\mu|$ holds, with a specialized Abelian case simplifying to $|a_{\mu_0}|\le\sum_{\mu\neq\mu_0}|a_\mu|$ and the explicit $(1+\tan(\theta/2))^n\ge 2$ criterion. The work further connects conclusive exclusion to zero-error communication, proving a lower bound $C_{0,F}(\mathcal{N})=C_{0,NS}(\mathcal{N})\ge \log\frac{|G|}{|G|-1}$ for group-generated classical-quantum channels via the fractional packing framework. Overall, the results illuminate how group symmetry constrains information extraction and have implications for quantum foundations and zero-error communication theory.

Abstract

Retrieving classical information from quantum systems is central to quantum information processing. As a more general task than quantum state discrimination, which focuses on identifying the exact state, quantum state exclusion only requires ruling out options, revealing fundamental limits of information extraction from quantum systems. In this work, we study the conclusive exclusion of quantum states generated by group actions, establishing explicit criteria for when such exclusion is possible. For systems with complex symmetries, including finite and compact Lie groups, we derive a sufficient condition for conclusive exclusion based on the initial state's amplitudes and the group's structure. As applications to special groups such as Abelian groups, we establish necessary and sufficient conditions for conclusive state exclusion and generalize the Pusey-Barrett-Rudolph result to a wider range of scenarios. Finally, we explore zero-error communication via conclusive exclusion of quantum states and derive a lower bound on the feedback-assisted and non-signalling-assisted zero-error capacity of classical-quantum channels generated by group actions.

Conclusive exclusion of quantum states with group action

TL;DR

This work analyzes conclusive single-state exclusion under group actions, linking symmetry to fundamental limits on quantum information extraction. By leveraging the isotypical decomposition of the representation and seed-state amplitudes, it derives a general sufficient condition for conclusive exclusion applicable to finite and compact Lie groups, and a tight necessary-and-sufficient condition for Abelian groups that recovers a generalized PBR result. A constructive POVM is provided, enabling perfect exclusion when the amplitude inequality holds, with a specialized Abelian case simplifying to and the explicit criterion. The work further connects conclusive exclusion to zero-error communication, proving a lower bound for group-generated classical-quantum channels via the fractional packing framework. Overall, the results illuminate how group symmetry constrains information extraction and have implications for quantum foundations and zero-error communication theory.

Abstract

Retrieving classical information from quantum systems is central to quantum information processing. As a more general task than quantum state discrimination, which focuses on identifying the exact state, quantum state exclusion only requires ruling out options, revealing fundamental limits of information extraction from quantum systems. In this work, we study the conclusive exclusion of quantum states generated by group actions, establishing explicit criteria for when such exclusion is possible. For systems with complex symmetries, including finite and compact Lie groups, we derive a sufficient condition for conclusive exclusion based on the initial state's amplitudes and the group's structure. As applications to special groups such as Abelian groups, we establish necessary and sufficient conditions for conclusive state exclusion and generalize the Pusey-Barrett-Rudolph result to a wider range of scenarios. Finally, we explore zero-error communication via conclusive exclusion of quantum states and derive a lower bound on the feedback-assisted and non-signalling-assisted zero-error capacity of classical-quantum channels generated by group actions.

Paper Structure

This paper contains 9 sections, 7 theorems, 27 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Main results
  4. Finite group and compact Lie group
  5. Generalized PBR Game
  6. Zero-error Communication
  7. Conclusions and discussions
  8. Elements of Representation Theory
  9. Zero-error communication

Key Result

Lemma 1

Let $\{M_g|g\in G\}$ be the POVM. Conclusive single-state exclusion of the group-orbit ${\cal E}_\psi$ can be done if and only if the operator $N:=\sum_{g\in G}|u_g\rangle\!\langle u_g|M_g$ is Hermitian and satisfies $N\leq|u_g\rangle\!\langle u_g|$ for all $g\in G$.

Figures (2)

  • Figure 1: The seed pure states with the form like Eq. \ref{['eq:input_state_u3']} that can be used to perfectly exclude states in ${\cal E}^{{\cal H}{\cal R}}_\psi$. Each arrow in this two diagrams represents a vector $a_\mu d_\mu=|a_\mu|d_\mu e^{i\theta_\mu}.$
  • Figure 2: Classical communication about finite group information.

Theorems & Definitions (8)

  • Lemma 1: bandyopadhyay2014conclusive
  • Theorem 2
  • Proposition 3
  • Corollary 4: Recover the condition of PBR game pusey2012reality
  • Proposition 5
  • Definition S1: Unitary Representation
  • Lemma S1: Schur lemma
  • Theorem S2: Commutant