Tang's ℓ2-Norm Sample-and-Query Model as the Unique Classical Implementation of the Born Rule from Five Operational Axioms

Abstract

Tang’s dequantisation programme established that several quantum machine learning algorithms admit efficient classical simulation when given ℓ2-norm sample-and-query (SQ) access to their input data. The specific choice of ℓ2-norm sampling—rather than uniform, ℓ1, or any other sampling rule—has hitherto been justified operationally by its polynomial-time implementability, but not axiomatically. We show that ℓ2-norm SQ sampling is the unique sampling functional satisfying five natural operational axioms (normalisation and positivity, additivity under coarse-graining, noncontextuality, unitary covariance, and continuity). The proof is direct and does not invoke Gleason’s theorem.This note operates top-down: it begins from the standard quantum formalism (Hilbert spaces, unitaries, density operators) and works downward to show that the five axioms uniquely force the ℓ2-norm sampling rule at the Instrumental layer of the Unified Emergence of Information and Reality (UEIR) operational pipeline [15]. It therefore provides a strong contribution to Criterion #2 (axiomatic justification for the Born rule) and Criterion #3 (the classical shadow / SQ access rule is the unique classical implementation of Born-rule measurement) of the quantum reconstruction programme. It does not derive Hilbert-spacestructure from first principles (Criterion #1), which remains an open direction stated explicitly as Open Problem 2.The same five axioms underlie the operational derivations of the Born rule in quantum mechanics (Hardy [7]; Chiribella–D’Ariano–Perinotti [8]; Masanes–M¨uller [9]). Tang’s SQ access is therefore not a design choice but the unique classical implementation of Born-rulemeasurement.