bibkey: marajaeizhang2026generalizedv3
authors: Haobo Ma, Reza Rajaei, Wenlin Zhang
year: 2026
title: "Generalized Fidelity, the Data Processing Inequality, and Convexity"
doi: null
url: https://arxiv.org/html/2609.09753v3
claim: "Version 3 supplies the ordered-root and qubit identities, a qubit DPI violation, and scalar sufficient criteria; the transported-midpoint commuting-input family and its outside-balance Good countermodels are treated separately here."
strata_touched:
  - D5/S3/Quantum/CommutingMidpoint
license: citation-only
triage: anchor

Generalized Fidelity, the Data Processing Inequality, and Convexity

This note cites arXiv:2609.09753v3 (20 October 2026), by Haobo Ma, Reza Rajaei and Wenlin Zhang. It does not replace the repository’s historical version-1 note or change the attribution of frozen suppliers.

The introduction, equations (1)-(3), fixes the ordered generalized fidelity and the generalized squared Bures quantity. Equation (3) is the weighted-root Hilbert-Schmidt norm formula and permits positive semidefinite input states. The reference is positive definite. The Qubit Case, Lemma 1, Lemmas 2-5 and Corollary 4 supply the two-dimensional root, Pauli/Bloch and fidelity quotient identities used by the current formalization. These inherited identities are credited to this source rather than claimed as new results.

Theorem 3 gives a qubit DPI violation. Thus the unresolved unrestricted qubit question in version 1 is historical, not the status in version 3. Corollary 7 gives scalar balance as a sufficient condition for every actual qubit channel with positive definite transported reference. The contribution history attributes the qubit formula and first two criteria to Rajaei, and the qubit counterexamples and stronger Corollary 7 to Ma and Zhang.

The current CommutingMidpoint candidate proves a positive family with commuting qubit density inputs, scalar input reference I/2, arbitrary complex qubit outputs, and positive definite transported reference. Its faithful and singular outside-balance triples satisfy Good over every admitted canonical channel. This is a negative statement about scalar-balance necessity, not a full replacement classification, a fixed-pair convexity result, or a worldwide priority claim.

Afham and Ferrie, arXiv:2410.04937v2, define the generalized-fidelity, squared-Bures and transported-reference conventions. Vuong, arXiv:2605.28885v1, recomputes pair-dependent polar references; those output references need not be the transported midpoint Phi(I/2). Neither source is used as an assumed transported-reference channel-DPI bridge.

Verified locator

Source: https://arxiv.org/html/2609.09753v3

Version: arXiv:2609.09753v3, 20 October 2026.

Verified HTML snapshot SHA-256: 485bdca9ed47570ebdcb59862c822c785a7d6b4a48740c353b08c2be56d229ca.