Petz recovery map

In Quantum information theory, the Petz recovery map is a quantum map that proposed by Dénes Petz^[1]. 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 }(\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 ${\mathcal {E}}$ .

References

^ Petz, Dénes (1986-03-01). "Sufficient subalgebras and the relative entropy of states of a von Neumann algebra". Communications in Mathematical Physics. 105 (1): 123–131. doi:10.1007/BF01212345. ISSN 1432-0916.

[1] Petz, Dénes (1986-03-01). "Sufficient subalgebras and the relative entropy of states of a von Neumann algebra". Communications in Mathematical Physics. 105 (1): 123–131. doi:10.1007/BF01212345. ISSN 1432-0916.

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