slug: rajaei-qubit-generalized-fidelity-dpi bibkey: rajaei2026generalized doi: null url: https://arxiv.org/html/2609.09753v1 triage: theorem motivation_gids:

• D5/S3/Quantum/GeneralizedFidelity.claim
• D5/S3/Quantum/FiniteDimensional
• D5/S3/Quantum/PointerBasis
• D5/S3/Quantum/Foundation/FiniteKrausChannel

Unrestricted qubit generalized-fidelity data processing

Problem

Rajaei’s abstract says: “In dimension two, although we do not settle the DPI in full generality,”. The exact residual assertion is that every trace-one positive semidefinite $P,Q\in M_2(\mathbb{C})$, every positive-definite trace-one $R$, and every completely positive trace-preserving channel $\Phi :M_2(\mathbb{C})\to M_2(\mathbb{C})$ with positive-definite $\Phi (R)$ obey

$$B_{\Phi (R)}(\Phi (P),\Phi (Q))\le B_R(P,Q).$$

Here $F_R(P,Q)$ is the ordered matrix-root trace in the source, and $B_R(P,Q)=\mathrm{Re}\mathrm{tr}(P+Q)-2\mathrm{Re}{F}_R(P,Q)$. Singular input states are permitted. Positivity only on three chosen states does not establish complete positivity.

Motivation

This is the paper’s residual dimension-two yes-or-no question, rather than its already refuted dimension-at-least-three statement or classification of all reference bases. It is a Tier-1 family target preregistered in https://github.com/the-omega-institute/trureturing/issues/13116.

Gap

GeneralizedFidelity.result is the closed negation of the full universal assertion, with no term premises. Its proof establishes a counterexample for every real $0<\epsilon \le 1/1000$ before choosing $\epsilon =1/1000$ for the final contradiction. The whole interval certificate is proof-local, not a separately exported family theorem or family GID. The source definitions use actual complex matrices, PSD roots, matrix inverses and the existing all-amplification completely positive channel.

Route

Use

$$P=\frac{1}{10}\left(\begin{array}{cc}1& 3\\ 3& 9\end{array}\right),Q=\frac{1}{5}\left(\begin{array}{cc}4& 2\\ 2& 1\end{array}\right),R_{\epsilon }=\mathrm{diag}(1-\epsilon ,\epsilon ),\Phi (M)=\frac{31}{32}M+\frac{1}{32}XMX.$$

The channel is independent of $\epsilon$. Its Kraus matrices are $\sqrt{31/32}I$ and $\sqrt{1/32}X$; their normalization feeds the existing all-amplification CPTP theorem. Its action equals the existing Fourier phase-damping map at retention $15/16$ for all complex matrices. The transported reference is $\mathrm{diag}((31-30\epsilon )/32,(1+30\epsilon )/32)$.

Direct positive-root certificates for both references and all four sandwiches recover the actual source trace. Each candidate is positive semidefinite and has the required square, so CFC.sqrt_unique identifies the actual matrix root. Positive diagonal entries certify reference invertibility and the displayed diagonal inverse. The sandwich determinant roots are zero at the input, and $S/1280$ and $3S/5120$ at the output. The root denominators have positive squared values $p=(1+8\epsilon )/10$, $q=(4-3\epsilon )/5$ at the input and $A,B$ at the output. The input formula is

$$F_{\mathrm{in}}=\frac{2+\epsilon }{\sqrt{2(4+29\epsilon -24{\epsilon }^2)}}>7/10.$$

For the output put

$$S=\sqrt{961+27900\epsilon -27900{\epsilon }^2},A=(95+450\epsilon +S)/640,B=(1955-1350\epsilon +3S)/2560.$$

The bounds $31\le S<32$ and $S\le 31+450\epsilon$ imply $63/320\le A\le 1/4$ and $3/4\le B\le 4/5$. With $C=893/1600$, the exact rectangle certificate is

$$(A+B-707/1600{)}^2-49AB/25\le -27/40000.$$

The positive numerator and denominators give $F_{\mathrm{out}}<7/10$. Trace-one proofs for the actual input and transported states justify reducing the source trace term to two, and imply $B_{\mathrm{in}}<3/5<B_{\mathrm{out}}$ throughout the interval. The final negation uses $\epsilon =1/1000$ only after this whole-interval certificate. No scalar proxy replaces the ordered matrix-root definition.

Falsifier

A root, inverse, positivity, normalization, channel-representation or interval gap prevents settlement. A scalar replacement for the source fidelity, one numerical witness alone, or hidden premises in the result would change the target. An equivalent earlier solution or current exact owner retires the candidate without a solved-problem increment.

Evidence

The closed Lean theorem GeneralizedFidelity.result negates the complete universal qubit assertion. Its proof-local quantified certificate covers every real $0<\epsilon \le 1/1000$ using actual PSD-root uniqueness, the normalized finite-Kraus channel and the strict source inequalities $B_{\mathrm{in}}<3/5<B_{\mathrm{out}}$. The rectangle bound is an exact real polynomial inequality; no finite numerical scan substitutes for the interval proof. The canonical Blueprint describes the four source definitions and this single authored theorem. Its Refuted resolution claim binds this dossier to the exact frozen result, whose type is the closed negation of the source-faithful claim.

Triage

Tier 1; resolution Refuted by D5/S3/Quantum/GeneralizedFidelity.result. The declaration retains proof_shape: bind-only and escape_witness: bind-only; admission_basis: open-problem-resolution is the published-problem exception under preregistration #13116. The computational utility is certified-instance with basis: refutes, and the typed result is Not claim. No companion theorem or additional mathematical declaration is part of this resolution.

The inherited bounded qualification inspected Rajaei v1, Afham–Ferrie v2, Vuong v1 and the indexed Afham citing work arXiv:2608.15833, together with 2017 GitHub issue titles/bodies, 623 problem records and obtainable public formal-library sources. It found no equivalent resolution or exact owner.

The proved mechanism is a strict reversal of the source’s normalized three-state quantity under a fixed, unital bit-flip channel. The source expresses this quantity through three pairwise Uhlmann fidelities; their ordinary data processing does not imply monotonicity of this normalization. The source’s two sufficient qubit conditions and its higher-dimensional results are separate assertions. Reference-base classification remains outside this target. Registration is paused; no completed information-escape audit is claimed.

ASSUMED-UNVERIFIED

Complete multi-index and global prior-solution exclusion remain unverified. Semantic Scholar and OpenAlex returned HTTP 429, and the bounded qualification cannot exclude unavailable or undiscoverable solutions. No worldwide priority claim follows.