Notes on l2 Norm of Coherence

Wei-Ning Li, Ming Fang, Yuan-Hong Tao +2 more

We investigate the relationships among the wave property, particle property and mixedness of quantum states based on the lp-norm coherence. By conforming that the lp-norm coherence is an appropriate measure of wave property and introducing a measure of particle property based on the differences between the maximal l2-norm coherence and the general l2-norm coherence, we present tradeoff relationships among the wave, particle and mixedness of quantum states. For 1<= p <2, we establish two kinds of tradeoffs of the wave, particle and mixedness with respect to the upper and lower bounds of the lp-norm coherence given by the l2-norm coherence. These trade relations give rise to compressive understanding of the intrinsic connections among the wave, particle and mixedness of quantum states, and cover some existing results as particular ones.

Piyu Li, Feng Liu, Yan-Qin Xu +1 more

A. Blanc-Lapierre, P. Dumontet

Tajudeen H. Sikiru

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.

Tajudeen H. Sikiru

We extend the operational uniqueness result of the ℓ2-norm sample-and-query rule [1] to the full quantum reconstruction programme. Starting from the five operational axioms A1–A5 of the companion note, we generalise them to the framework of generalised probabilistic theories (GPTs). We impose two additional structural postulates: the strict non-commutative UEIR ordering R ◦ I ◦ E ◦ U and the conductance geometry Φ = k/n derived from the Boolean hypercube Qn.We prove (Theorem 5.1) that the only GPT satisfying A1–A5 together with the UEIR ordering and the conductance geometry is complex Hilbert-space quantum mechanics (d ≥ 2). As corollaries we recover the standard Born rule p(k|ρ) = Tr(ρEk) on density operators and the emergence of Tang’s ℓ2-norm SQ access model as the unique R-layer classical realisation.We further prove (Theorem 7.1) that under Born-rule ℓ2-norm sampling, the expected conductance of Haar-random quantum states satisfies EHaar[Φeff ] = (n + 1)/(2n) → 1/2 as n → ∞. This makes the quantum-classical threshold pc = 1/2 a proved theorem derivable from quantum mechanics, not an external postulate. We identify a structural parallel between this convergence and the Riemann critical line Re(s) = 1/2, arising from the shared involution structure of L2-norm squaring, and state a precise open conjecture.

L. E. J. Brouwer