Why is it called the Petz 'transpose' map?

Question

The Petz transpose map (also called Petz recovery map) is defined for a Channel $\Phi : \mathrm{L}(\mathcal X) \to \mathrm{L}(\mathcal Y) $ and an input reference state $\rho \in \mathrm{D}(\mathcal X )$ as $$ \mathcal R_{\Phi, \rho} (Y) := \rho ^{\frac12} \left[\Phi^* (\Phi(\rho)^{-\frac12} Y \Phi(\rho)^{-\frac12} ) \right] \rho ^{\frac12}, $$ for any $Y \in \mathrm{L}(\mathcal Y)$. Here $\Phi^* : \mathrm{L}(\mathcal Y) \to \mathrm{L}(\mathcal X) $ is the adjoint map of $\Phi$.

I was wondering why the map is called the 'Transpose' map, as a direct inspection does not reveal any tranpose operations involved.

Answer 1

It's usually called Petz recovery map as it reverses the effect of the quantum channel $\Psi$ on a state $\rho$. I guess it's sometimes called the "tranpose" map because it involves the adjoint of the channel, which in functional analysis can be viewed as a generalization of the matrix transpose operation.