On the relevance of Hermitian-preserving trace-preserving linear maps in asymptotic information retrieval in quantum communication

Abstract

In quantum technologies, quantum channels are essential for transmitting quantum states. Quantum channels usually introduce noise, thereby reducing the information in a state. 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. Precisely, given a pair of quantum channels ${\Lambda }_1,{\Lambda }_2$ and an arbitrary unknown input state, we show that the output state of the quantum channel ${\Lambda }_2$ from the pair can be uniquely identified from the output statistics of the other channel ${\Lambda }_1$ from the pair, using some POVM, iff ${\Lambda }_2$ can be obtained from ${\Lambda }_1$ by concatenating it with a Hermitian-preserving trace-preserving linear map that is not necessarily positive. In such cases, ${\Lambda }_2$ may not always be obtained from ${\Lambda }_1$ through post-processing. The relation between these channels is a preorder, and we study its characterization in this work. In other words, we provide an operational interpretation of this preorder induced by HPTP linear maps in a communication scenario. Furthermore, we characterize the difficulty of implementing ${\Lambda }_2$ given that ${\Lambda }_1$ 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 provide valuable insights about the relevance of Hermitian-preserving trace-preserving linear maps in physically motivated settings.

arXiv comment14 pages, 2 figures, Typos fixed