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. These are mathematically described by completely positive trace-preserving linear maps that represent the generic evolution of quantum systems and are also special cases of Hermitian-preserving trace-preserving (HPTP) linear maps. Concatenating a quantum channel with another quantum channel makes it noisier and degrades its information and resource preservability. 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 uniquely identified from the output statistics of the other channel from the pair using some quantum measurement, then the former channel from the pair can be obtained from the latter channel by concatenating it with a Hermitian-preserving trace-preserving linear map that is not necessarily positive. In such cases, the former channel may not always be obtained from the latter through post-processing. This relation between these two channels is a preorder, and we try to study its characterization in this work. Furthermore, we try to characterize the difficulty of implementing the former channel given that the latter 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. In short, we try to provide valuable insights about the relevance of Hermitian-preserving trace-preserving linear maps in physically motivated settings.

💡 Research Summary

The manuscript investigates a novel way of comparing quantum channels by allowing the intermediate map that connects them to be any Hermitian‑preserving trace‑preserving (HPTP) linear map, rather than restricting it to completely positive trace‑preserving (CPTP) maps. The authors start by recalling the standard formalism of quantum measurements, channels, and instruments, and they emphasize that CPTP maps are a proper subset of HPTP maps (CPTP ⊂ CP ⊂ HPTP). They then introduce informationally complete (IC) measurements, showing that such measurements uniquely determine an unknown state and that a minimal IC measurement provides a linear, invertible relation between outcome probabilities and the state’s expansion coefficients.

The core contribution is the definition of an “asymptotic power” preorder, denoted Λ₁ ⪰ₐₛᵧₘₚ Λ₂. Two channels satisfy this relation if there exist IC measurements M₁ on the output of Λ₁ and M₂ on the output of Λ₂ (with identical outcome sets) such that for every input state ρ the outcome statistics coincide: Tr