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The swap transpose on couplings translates to Petz' recovery map on quantum channels
Gergely Bunth, József Pitrik, Tamás Titkos, Dániel Virosztek·December 4, 2025
Mathematical Physicsmath.OAQuantum Physics
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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 $Π_Φ$ corresponding to the channel $Φ$ and initial state $ρ$ is the coupling $Π_{rec}$ corresponding to the Petz recovery map $Φ_{rec}.$