Comparing quantum channels using Hermitian-preserving trace-preserving linear maps: A physically meaningful approach

In quantum technologies, quantum channels are essential elements for the transmission of quantum states. The action of a quantum channel usually introduces noise in the quantum state and thereby reduces the information contained in it. Concatenating a quantum channel with another quantum channel makes it more noisy and degrades its information and resource preservability. These are mathematically described by completely positive trace-preserving linear maps that represent the generic evolution of quantum systems. These are special cases of Hermitian-preserving trace-preserving linear maps. In this work, we demonstrate a physically meaningful way to compare a pair of quantum channels using Hermitian-preserving trace-preserving linear maps. More precisely, given a pair of quantum channels and an arbitrary unknown input state, we show that if the output state of one quantum channel from the pair can be obtained from the output statistics of the other channel from the pair using some quantum measurement, then the latter channel from the pair can be obtained from the former channel by concatenating it with a Hermitian-preserving trace-preserving linear map. This relation between these two channels is a preorder, and we try to study its characterization in this work. We try to characterize the difficulty of implementing the latter channel, given that the former channel has already been implemented via a quantifier, namely, physical implementability. We also illustrate the implications of our results for the incompatibility of quantum devices through an example.