For minimum-error channel discrimination tasks that involve only unitary channels, we show that sequential strategies may outperform the parallel ones. Additionally, we show that general strategies that involve indefinite causal order are also advantageous for this task. However, for the task of discriminating a uniformly distributed set of unitary channels that forms a group, we show that parallel strategies are, indeed, optimal, even when compared to general strategies. We also show that strategies based on the quantum switch cannot outperform sequential strategies in the discrimination of unitary channels. Finally, we derive an absolute upper bound for the maximal probability of successfully discriminating any set of unitary channels with any number of copies for the most general strategies that are suitable for channel discrimination. Our bound is tight since it is saturated by sets of unitary channels forming a group k-design.
Here, stands for the inverse of on its range. If the operator W is not not full rank, the composition W W−1 ≕ ΠW is not the identity but the projector onto the subspace spanned by the range of . Due to this technicality, when the operator W is not full rank, we should define the measurements as . With that, the proof written here also applies to the case where the operator W is not full rank.
Indeed, two diagonalisable operators A and B commute if and only if they are diagonal in the same basis. Now, if A ≕ ∑iαi|i〉〈i|, its positive semidefinite square root is also diagonal in the same basis, in particular, . Hence, if A commutes with B, also commutes with B.
Interestingly, it can be proved that for the k = 2-slot case, the action of the standard quantum switch superchannel, i.e., , on unitary channels uniquely defines its action on general operations.61 The possibility of extending this result for general switch-like superchannels is still open, but the existence of general switch-like superchannels is ensured by the construction presented via Eq. (E12).