Conclusive exclusion of quantum states with group action

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 conclusive state exclusion of quantum states under group actions. First, we establish explicit criteria for achieving conclusive exclusion for states under group actions. For complex symmetries, including finite groups and compact Lie groups, we derive a sufficient condition for exclusive exclusion based solely on amplitudes of the seed state and group structure parameters. Second, 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. Third, 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.