Petz recovery map

In quantum information theory, a mix of quantum mechanics and information theory, the Petz recovery map can be thought of a quantum analog of Bayes theorem. Proposed by Dénes Petz, the Petz recovery map is a quantum channel associated with a given quantum channel and quantum state. This recovery map is designed in a manner that, when applied to an output state resulting from the given quantum channel acting on an input state, it enables the inference of the original input state. In essence, the Petz recovery map serves as a tool for reconstructing information about the initial quantum state from its transformed counterpart under the influence of the specified quantum channel.

The Petz recovery map finds applications in various domains, including quantum retrodiction, quantum error correction, and entanglement wedge reconstruction for black hole physics.

Definition
Suppose we have a quantum state which is described by a density operator $$\sigma$$ and a quantum channel $$\mathcal{E}$$, the Petz recovery map is defined as


 * $$\mathcal{P}_{\sigma,\mathcal{E}}(\rho)=\sigma^{1/2}\mathcal{E}^{\dagger}(\mathcal{E}(\sigma)^{-1/2}\rho \mathcal{E}(\sigma)^{-1/2})\sigma^{1/2}.$$

Notice that $$\mathcal{E}^{\dagger}$$is the Hilbert-Schmidt adjoint of $$\mathcal{E}$$.

The Petz map has been generalized in various ways in the field of quantum information theory.

Properties of the Petz recovery map
$$\begin{aligned} \operatorname{Tr}\left[\mathcal{P}_{\sigma, \mathcal{N}}(X)\right] & =\operatorname{Tr}\left[\sigma^{\frac{1}{2}} \mathcal{E}^{\dagger}\left(\mathcal{E}(\sigma)^{-\frac{1}{2}} X \mathcal{E}(\sigma)^{-\frac{1}{2}}\right) \sigma^{\frac{1}{2}}\right] \\ & =\operatorname{Tr}\left[\sigma \mathcal{E}^{\dagger}\left(\mathcal{E}(\sigma)^{-\frac{1}{2}} X \mathcal{E}(\sigma)^{-\frac{1}{2}}\right)\right] \\ & =\operatorname{Tr}\left[\mathcal{E}(\sigma) \mathcal{E}(\sigma)^{-\frac{1}{2}} X \mathcal{E}(\sigma)^{-\frac{1}{2}}\right] \\ & =\operatorname{Tr}\left[\mathcal{E}(\sigma)^{-\frac{1}{2}} \mathcal{E}(\sigma) \mathcal{E}(\sigma)^{-\frac{1}{2}} X\right] \\ & =\operatorname{Tr}\left[\Pi_{\mathcal{E}(\sigma)} X\right] \\ & \leq \operatorname{Tr}[X] \end{aligned} $$ $$ is invertable, the Petz recovery map $$\mathcal{P}_{\sigma,\mathcal{E}}$$ is a quantum channel, viz., a completely positive trace-preserving (CPTP) map.
 * 1) The Petz recovery map is a completely positive map, since (i) sandwiching by the positive semi-definite operator $$\mathcal{E}(\sigma)^{-1/2}(\cdot) \mathcal{E}(\sigma)^{-1/2}$$is completely positive; (ii) $$\mathcal{E}^{\dagger}$$is also completely positive when  $$\mathcal{E}$$ is completely positive; and (iii) sandwiching by the positive semi-definite operator $$\sigma^{1/2}(\cdot)\sigma^{1/2}$$ is completely positive.
 * 2) It's also clear that  $$\mathcal{P}_{\sigma,\mathcal{E}}$$ is  is trace non-increasing
 * 1) From 1 and 2, when $$\mathcal{E}(\sigma)