Operational interpretation of the Choi rank through exclusion tasks

The Choi state is an indispensable tool in the study and analysis of quantum channels. Considering a channel in terms of its associated Choi state can greatly simplify problems. It also offers an alternative approach to the characterization of a channel, with properties of the Choi state providing novel insight into a channel's behavior. The rank of a Choi state, termed the Choi rank, has proven to be an important characterizing property, and here, its significance is further elucidated through an operational interpretation. The Choi rank is shown to provide a universal bound on how successfully two agents, Alice and Bob, can perform an entanglement-assisted exclusion task. The task can be considered an extension of superdense coding, where Bob can only output information about Alice's encoded bit string with certainty. Conclusive state exclusion, in place of state discrimination, is therefore considered at the culmination of the superdense coding protocol. In order to prove this result, a necessary condition for conclusive <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>k</a:mi></a:math>-state exclusion of a set of states is presented in order to achieve this result, and the notions of weak and strong exclusion are introduced. Published by the American Physical Society 2024