Unitary channel discrimination beyond group structures: Advantages of sequential and indefinite-causal-order strategies

Here, $W−1$ stands for the inverse of $W$ on its range. If the operator W is not not full rank, the composition W W⁻¹ ≕ Π_W is not the identity $1$ but the projector onto the subspace spanned by the range of $W$⁠. Due to this technicality, when the operator W is not full rank, we should define the measurements as $MUI′O:=W−1I′OTUI′OW−1I′O+1N(1−WW−1)$⁠. 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 ≕ ∑_iα_i|i〉〈i|, its positive semidefinite square root is also diagonal in the same basis, in particular, $A=∑iαi|i〉〈i|$⁠. Hence, if A commutes with B, $A$ 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., $Vπn=1$⁠, 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).