The swap transpose on couplings translates to Petz' recovery map on quantum channels

TL;DR

This paper establishes a precise correspondence between the swap transpose operation on quantum couplings and the Petz recovery map on quantum channels, linking two concepts in quantum information theory.

Contribution

It demonstrates that the Petz recovery map is the coupling counterpart of the swap transpose operation on quantum couplings, clarifying their relationship.

Findings

Swap transpose of couplings corresponds to Petz recovery map.

Provides a one-to-one correspondence between couplings and channels.

Enhances understanding of quantum transport and recovery processes.

Abstract

In [Ann. Henri Poincar\'e, {\bf 22} (2021), 3199-3234], De Palma and Trevisan described a one-to-one correspondence between quantum couplings and quantum channels realizing transport between states. The aim of this short note is to demonstrate that taking the Petz recovery map for a given channel and initial state is precisely the counterpart of the swap transpose operation on couplings. That is, the swap transpose of the coupling $\Pi_{\Phi}$ corresponding to the channel $\Phi$ and initial state $\rho$ is the coupling $\Pi_{rec}$ corresponding to the Petz recovery map $\Phi_{rec}.$