Ordinary-Petz recovery bound for conditional mutual information

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

\begin{equation} I(A;B\mid C)_\rho :=S(AC)_\rho+S(BC)_\rho-S(C)_\rho-S(ABC)_\rho, \qquad S(\tau):=-\operatorname{Tr}(\tau\log_2\tau). \tag{1} \end{equation}

The ordinary Petz map associated with \(\rho_{AC}\) and the channel \(\operatorname{Tr}_A:AC\to C\) is

\begin{equation} \mathcal P^{\rho}_{C\to AC}(X_C) :=\rho_{AC}^{1/2}\!\left[ I_A\otimes\rho_C^{-1/2}X_C\rho_C^{-1/2} \right]\rho_{AC}^{1/2}, \tag{2} \end{equation}

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

\begin{equation} I(A;B\mid C)_\rho \stackrel{?}{\geq} -\log_2 F\!\left( \rho_{ABC}, (\operatorname{id}_B\otimes\mathcal P^{\rho}_{C\to AC})(\rho_{BC}) \right), \tag{3} \end{equation}

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

Peter supplied two explicit three-qubit counterexamples to Eq. (3) through the project submission inbox on 10 September 2026. For the rank-three example, in computational \(ABC\) order, set

\begin{equation} \begin{aligned} v_1&=4\lvert111\rangle,\qquad v_2=3(\lvert011\rangle-\lvert101\rangle),\\ v_3&=5\lvert001\rangle+\lvert010\rangle+\lvert100\rangle,\qquad \rho_{ABC}=\frac1{61}\sum_{j=1}^3\lvert v_j\rangle\!\langle v_j\rvert. \end{aligned} \tag{4} \end{equation}

The vectors in Eq. (4) are orthogonal with squared norms \(16,18,27\), so they define a rank-three density operator. Its marginals satisfy

\begin{equation} \rho_{AC}=\rho_{BC}=\frac1{61} \begin{pmatrix}1&0&0&0\\0&34&5&0\\0&5&1&0\\0&0&0&25\end{pmatrix}, \qquad \rho_C=\frac1{61}\operatorname{diag}(2,59). \tag{5} \end{equation}

Thus the inverse in Eq. (2) is an ordinary inverse. For \(\sigma_{ABC}=(\operatorname{id}_B\otimes\mathcal P^{\rho}_{C\to AC})(\rho_{BC})\), independent exact symbolic reconstruction and rational interval arithmetic give

\begin{equation} \begin{aligned} I(A;B\mid C)_\rho&=0.48665355975796833566\ldots,\\ -\log_2F(\rho_{ABC},\sigma_{ABC})&=0.49106481978485687802\ldots,\\ -0.004412&<I(A;B\mid C)_\rho+\log_2F(\rho_{ABC},\sigma_{ABC})<-0.0044<0. \end{aligned} \tag{6} \end{equation}

Equation (6) disproves the universal inequality with exactly the map and squared-fidelity convention in the statement. A second, rank-two example has a separately certified gap between \(-0.000881\) and \(-0.000880\). Both constructions, their exact fidelity matrices, and an independently written verifier are recorded in the counterexample note [Pet26].
A structurally distinct classical-flag construction gives a third three-qubit counterexample. Let \(V|0\rangle=|00\rangle_{AC}\) and \(V|1\rangle=|11\rangle_{AC}\), and define

\begin{equation} \sigma=\begin{pmatrix}1/2000&-1/75\\-1/75&1999/2000\end{pmatrix}, \qquad \rho=\begin{pmatrix}3/4&\sqrt3/4\\\sqrt3/4&1/4\end{pmatrix}, \qquad p=\frac{3000}{3001},\quad q=\frac1{3001}. \tag{7} \end{equation}

Using the data in Eq. (7), in canonical \(ABC\) order, set

\begin{equation} \omega_{ABC}=p|0\rangle\!\langle0|_B\otimes V\sigma V^\dagger +q|1\rangle\!\langle1|_B\otimes V\rho V^\dagger. \tag{8} \end{equation}

Exact reduction of Eq. (8), followed by directed Arb ball arithmetic, certifies

\begin{equation} I(A;B\mid C)_\omega<\frac{435}{10^6} <\frac{466}{10^6} <-\log_2 F\!\left(\omega_{ABC}, (\operatorname{id}_B\otimes\mathcal P^\omega_{C\to AC})(\omega_{BC}) \right). \tag{9} \end{equation}

Thus Eq. (9) independently disproves the same ordinary-map inequality. The analytic reduction, interval verifier, and a full-\(8\times8\)-matrix audit are recorded in the flagged-state counterexample note [QOP26].
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{10} \end{equation}

for an explicitly averaged recovery map \(\mathcal R_{C\to AC}\). Equation (10) 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. (10). 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 ordinary-map requirement in Eq. (2) is settled negatively by the compatible-marginal counterexamples above. The results using rotations, pinching, averaging, or optimized recovery channels remain valid. Peter’s two submitted examples and the independent flagged-state construction are verified with rigorous interval methods; neither result has external peer review or a proof-assistant kernel check, and no historical-priority claim is asserted [Pet26], [QOP26].

References

[BSW15]: M. Berta, K. P. Seshadreesan, and M. M. Wilde, “Rényi Generalizations of the Conditional Quantum Mutual Information,” Journal of Mathematical Physics 56, 022205 (2015).DOI arXiv
[Wil17]: M. M. Wilde, Quantum Information Theory, 2nd ed., Cambridge University Press (2017), Sec. 12.7.DOI arXiv
[STH16]: D. Sutter, M. Tomamichel, and A. W. Harrow, “Strengthened Monotonicity of Relative Entropy via Pinched Petz Recovery Map,” IEEE Transactions on Information Theory 62, 2907–2913 (2016).DOI arXiv
[JRS+18]: M. Junge, R. Renner, D. Sutter, M. M. Wilde, and A. Winter, “Universal Recovery Maps and Approximate Sufficiency of Quantum Relative Entropy,” Annales Henri Poincaré 19, 2955–2978 (2018).DOI arXiv
[Pet26]: Peter, “Two three-qubit counterexamples to the ordinary-Petz CMI bound,” project submissions (10 September 2026), consolidated and independently verified for QIQCOP Zoo (12 September 2026). Counterexample note and exact verification certificate.link
[QOP26]: A flagged-state three-qubit counterexample to the ordinary-Petz CMI bound, independent QIQCOP Zoo contribution (10 September 2026), verified 13 September 2026. Proof and verification package.link

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