bibkey: rajaei2026generalized authors: Reza Rajaei year: 2026 title: “Generalized Fidelity and the Data Processing Inequality” doi: null url: https://arxiv.org/html/2609.09753v1 claim: “Generalized Bures–Wasserstein data processing fails in dimensions at least three; the unrestricted qubit case remains unresolved, with two sufficient conditions proved.” strata_touched:

• D5/S3/Quantum/GeneralizedFidelity license: citation-only triage: anchor

Generalized Fidelity and the Data Processing Inequality

The abstract of arXiv:2609.09753v1 states:

In dimension two, although we do not settle the DPI in full generality,

The introduction defines

$$F_R(P,Q)=\mathrm{tr}\left(\sqrt{\sqrt{R}P\sqrt{R}}{R}^{-1}\sqrt{\sqrt{R}Q\sqrt{R}}\right),B_R(P,Q)=\mathrm{tr}(P+Q)-2\mathrm{Re}{F}_R(P,Q).$$

Every root is the positive semidefinite matrix root. The reference $R$ must be positive definite; the introduction explicitly permits rank-deficient $P,Q$. On density matrices, the trace term equals two. The residual qubit question asks whether $B_{\Phi (R)}(\Phi (P),\Phi (Q))\le B_R(P,Q)$ for every completely positive trace-preserving qubit channel with positive-definite $\Phi (R)$.

The section The Qubit Case derives the nonzero positive semidefinite two-dimensional root formula

$$\sqrt{T}=\frac{T+\sqrt{\det T}I}{\sqrt{\mathrm{tr}T+2\sqrt{\det T}}}.$$

It obtains an explicit real generalized-fidelity formula in terms of three pairwise Uhlmann fidelities and proves two sufficient conditions for data processing. These partial conditions do not settle the universal qubit question. The paper’s higher-dimensional counterexample is a different settlement. Its necessary and sufficient condition for equality with Uhlmann fidelity is also a separate statement.

Related sources are Afham and Ferrie, Riemannian-geometric generalizations of quantum fidelities and Bures-Wasserstein distance, arXiv:2410.04937v2, and Vuong, Polar Fidelities, Holevo Bases, and Unitary Factors of Generalized Fidelity, arXiv:2605.28885v1. The residual qubit universal inequality is narrower than classification of all admissible reference bases.

Verified locator

URL: https://arxiv.org/html/2609.09753v1 (version 1). The abstract explicitly retains the unrestricted dimension-two DPI question. The introduction defines the ordered matrix-root fidelity and the generalized Bures–Wasserstein quantity and permits rank-deficient input states. The Qubit Case gives the two-dimensional PSD-root formula and the two sufficient qubit DPI conditions.

Repository resolution

The unrestricted qubit assertion is Refuted by the frozen GeneralizedFidelity.result. This closed theorem negates the full universal assertion using the source’s actual ordered PSD-root trace and a normalized two-Kraus CPTP channel. Its proof establishes $B_{\mathrm{in}}<3/5<B_{\mathrm{out}}$ for every real $0<\epsilon \le 1/1000$ before selecting a witness for the negation. The source’s two sufficient qubit conditions remain separate results.

The resolution dossier is rajaei-qubit-generalized-fidelity-dpi. Admission uses open-problem-resolution under preregistration #13116, with proof_shape: bind-only. Global prior-solution exclusion remains ASSUMED-UNVERIFIED; no worldwide priority is claimed. Registration is paused and the information-escape audit is unfinished.