Ordinary-Petz fidelity remainder for relative-entropy data processing
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Problem
Does the ordinary Petz recovery map give a universal fidelity remainder for monotonicity of quantum relative entropy? Let \(\mathcal N:\mathcal L(A)\to\mathcal L(B)\) be a finite-dimensional quantum channel, and let \(\rho,\sigma\in\mathcal D(A)\) satisfy \(\operatorname{supp}\rho\subseteq\operatorname{supp}\sigma\). With inverses taken on the relevant supports, define the Petz map by
where \(\mathcal N^{\dagger}\) is the Hilbert–Schmidt adjoint. Write \(D(\tau\Vert\omega):=\operatorname{Tr}[\tau(\log_2\tau- \log_2\omega)]\) and use squared fidelity \(F(\tau,\omega):=\lVert\sqrt\tau\sqrt\omega\rVert_1^2\). The proposed remainder bound for the map in Eq. (1) is
Determine whether Eq. (2) holds for every such triple \((\rho,\sigma,\mathcal N)\).
Source
Seshadreesan, Berta, and Wilde explicitly proposed the monotonicity in the Rényi parameter whose endpoint consequence is Eq. (2); Wilde records the same conjectured ordinary-Petz remainder in Section 12.4 [SBW15], [Wil17].
Progress
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Bhattacharya disproved Eq. (2) using the diagonal pinching channel \(\Phi:M_2(\mathbb C)\to M_2(\mathbb C)\) and the density matrices
\begin{equation} A=\begin{pmatrix} \tfrac12&\tfrac12\\ \tfrac12&\tfrac12 \end{pmatrix}, \qquad B=\begin{pmatrix} \tfrac34&-\tfrac14\\ -\tfrac14&\tfrac14 \end{pmatrix}, \qquad \Phi(X)=\sum_{j=0}^{1}|j\rangle\!\langle j|X|j\rangle\!\langle j|. \tag{3} \end{equation}For \((\rho,\sigma,\mathcal N)=(A,B,\Phi)\) from Eq. (3), direct evaluation with natural logarithms and root fidelity gives a data-processing loss of approximately \(1.5191\), while the proposed recovery term is approximately \(1.5349\). Since replacing root fidelity by squared fidelity and natural logarithms by base-two logarithms rescales both sides consistently, this is also a counterexample to the convention in Eq. (2) [Bha25].
The counterexample in Eq. (3) also rules out the full Rényi-parameter monotonicity proposed by Seshadreesan, Berta, and Wilde, because that monotonicity implies the false endpoint inequality Eq. (2) [SBW15], [Bha25].
Comment
The answer to Eq. (2) is negative already in dimension two. This general-channel counterexample does not resolve the more structured conditional-mutual-information inequality in the ordinary-Petz conditional-mutual-information problem, where the channel is a partial trace and the reference state is a marginal of the state being recovered.