Ordinary-Petz recovery bound for conditional mutual information
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- Topics
Problem
Does the ordinary, unrotated Petz map universally recover a tripartite state with fidelity controlled by its conditional mutual information? For a finite-dimensional state \(\rho_{ABC}\), define
The ordinary Petz map associated with \(\rho_{AC}\) and the channel \(\operatorname{Tr}_A:AC\to C\) is
where the inverse is taken on \(\operatorname{supp}\rho_C\). With squared fidelity \(F(\tau,\omega):=\lVert\sqrt\tau\sqrt\omega\rVert_1^2\), determine whether the quantity in Eq. (1) always satisfies
with the recovered systems ordered canonically as \(ABC\).
Source
Berta, Seshadreesan, and Wilde state the Rényi-monotonicity conjecture whose \(\alpha=1/2\) and \(\alpha=1\) endpoints give Eq. (3); Wilde records this ordinary-Petz inequality explicitly in Section 12.7 [BSW15], [Wil17].
Progress
Sutter, Tomamichel, and Harrow proved a strengthened data-processing inequality using a pinched Petz map, equivalently a convex combination of rotated Petz maps. Their result yields a conditional-mutual-information recovery bound of the form
\begin{equation} I(A;B\mid C)_\rho \geq-\log_2 F\!\left( \rho_{ABC}, (\operatorname{id}_B\otimes\mathcal R_{C\to AC})(\rho_{BC}) \right), \tag{4} \end{equation}for an explicitly averaged recovery map \(\mathcal R_{C\to AC}\). Equation (4) does not establish Eq. (3), because the averaging need not reduce to the unrotated map in Eq. (2) [STH16].
Junge, Renner, Sutter, Wilde, and Winter constructed a universal recovery map depending only on the reference state and the channel, and proved a fidelity remainder of the form in Eq. (4). Their universal map is an average of rotated Petz maps, so universality alone does not settle the ordinary-map requirement in Eq. (3) [JRS+18].
Comment
The unresolved requirement is the specific map in Eq. (2), without rotations, pinching, averaging, or an optimization over recovery channels. The general data-processing counterexample recorded in Problem does not have the constrained partial-trace and compatible-marginal structure imposed here.