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

Abstract: In [Ann. Henri Poincaré, {\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 $ΠΦ$ corresponding to the channel $Φ$ and initial state $ρ$ is the coupling $Π{rec}$ corresponding to the Petz recovery map $Φ_{rec}.$