Operational Interpretation of the Choi Rank Through k-State Exclusion

TL;DR

This paper links the Choi-rank of quantum channels to an operational task, showing it bounds the success of entanglement-assisted exclusion, thus providing a new interpretative perspective on Choi-states.

Contribution

It introduces an operational interpretation of the Choi-rank by relating it to the success probability in a quantum exclusion task, extending understanding of Choi-states in quantum information.

Findings

Choi-rank bounds entanglement-assisted exclusion success

Operational interpretation of Choi-rank established

Necessary conditions for k-state exclusion derived

Abstract

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 characterisation of a channel, with properties of the Choi-state providing novel insight into a channel's behaviour. The rank of a Choi-state, termed the Choi-rank, has proven to be an important characterising 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 super-dense 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 super-dense coding protocol. In order to prove this result, a necessary condition for conclusive k-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.