Form≥2these fluctuations are small, so write the purity asTrρ 2 A =⟨Trρ 2 A⟩(1 +ϵ)withϵ= (Trρ2 A − ⟨Trρ2 A⟩)/⟨Trρ2 A⟩, so that⟨ϵ⟩= 0and⟨ϵ 2⟩= Var(Trρ2 A)/⟨Trρ2 A⟩2

The Jensen gap Since the mean purity is pinned, the entiremdependence of⟨S 2⟩is the gap between the mean of the logarithm, the logarithm of the mean—the Jensen gap, ⟨S2⟩=−ln T

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Typical Entanglement of Superpositions

quant-ph · 2026-07-07 · conditional · novelty 7.0

An m-fold superposition of typical sub-maximally entangled states gains a universal ln(m) entanglement enhancement, while maximally entangled states relax to the Haar limit via N-independent scaling laws.

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Typical Entanglement of Superpositions quant-ph · 2026-07-07 · conditional · none · ref 19
An m-fold superposition of typical sub-maximally entangled states gains a universal ln(m) entanglement enhancement, while maximally entangled states relax to the Haar limit via N-independent scaling laws.

Form≥2these fluctuations are small, so write the purity asTrρ 2 A =⟨Trρ 2 A⟩(1 +ϵ)withϵ= (Trρ2 A − ⟨Trρ2 A⟩)/⟨Trρ2 A⟩, so that⟨ϵ⟩= 0and⟨ϵ 2⟩= Var(Trρ2 A)/⟨Trρ2 A⟩2

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