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Sharp Scalar Fidelity Endpoint Comparison

Abstract

A sharp scalar condition characterizes dominance of the fidelity quotient over every independent upper endpoint in the unit cube.

Definition 1.1 (The scalar quotient).

Formalization. D5/S3/Quantum/ScalarFidelityEndpoint.scalarQuotient (✓ std3).

Citation. Reza Rajaei (2026). Generalized Fidelity and the Data Processing Inequality. URL: https://arxiv.org/html/2609.09753v1.

Commentary.

For real a,b,c, scalarQuotient(a,b,c), denoted by g(a,b,c), is (a squared+b squared+c squared-1)/(2ab). The theorem uses strictly positive a and b, so its denominators are positive. Rajaei’s qubit identity motivates this scalar expression; the scalar definition itself imposes no matrix realization.

Theorem 1.2 (Exact lower-corner dominance).

Proof. Machine-checked in Lean as D5/S3/Quantum/ScalarFidelityEndpoint.lower_corner_dominance_iff (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Reza Rajaei (2026). Generalized Fidelity and the Data Processing Inequality. URL: https://arxiv.org/html/2609.09753v1.

Commentary.

For every base point with 0<a,b<=1 and 0<=c<=1, the displayed equivalence includes both directions and all closed upper endpoints. The upper endpoints vary independently. Neither the endpoint nor any intervening point is required to satisfy the squared-imbalance condition. The sharp scalar argument is due to Haobo Ma and Wenlin Zhang in their corrected note, Monotonicity of the qubit fidelity quotient and fixed-reference convexity. Rajaei’s Corollaries 5 and 6 supply the earlier sufficient criteria.

For sufficiency put d=1-c squared, x=A-a and y=B-b. Clearing the positive denominators gives the nonnegative expression below. The condition bounds both signed squared differences by d and also bounds (a-b) squared by d. Increasing c to C adds a nonnegative contribution because 0<=c<=C.

For necessity, if b squared-a squared exceeds 1-c squared, choose s=sqrt(b squared+c squared-1). Then a<s<=1, and the difference after increasing a to s has numerator -bs(s-a) squared after multiplication by positive denominators. It is strictly negative. Interchanging a and b gives the other signed bound. This proves the sharpness at the lower corner, including c=0 and c=1. It does not characterize physical channel realizability, prove a matrix fidelity bridge, or classify CPTP maps.

References