Manuscript Em06-B@1 · Em06-B-v1.tex · proposition · 61e38859

Proposition: Intrinsic and measured concentration are different

One declared measurement benchmark and preparation cost › An additive-noise POVM, not an apparatus claim · lines 1613–1633 · proof included or linked

Em06-B@1#prop:noise, Em06-B@1#eq:measurement-budget

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For the POVM (eq:noise-POVM),

E ∣ Y_{ϵ} - o ∣^{2} = ϵ ⟨ Φ (v)^{2} ⟩_{ρ_{ϵ}} + σ_{ϵ}^{2} .

Measured $W_{2}$ concentration on $o$ holds exactly when both nonnegative terms tend to zero. In particular, fixed nonzero detector variance prevents measured concentration even when the intrinsic acceptance pair holds.

Proof. Expand $(X_{ϵ} - o + η_{ϵ})^{2}$ . Independence and the zero mean of $η_{ϵ}$ eliminate the cross term. The intrinsic term is (eq:mse). Equation (eq:pointmass) gives the Wasserstein assertion.

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Em06-B@1#eq:measurement-budget · line 1616
$E ∣ Y_{ϵ} - o ∣^{2} = ϵ ⟨ Φ (v)^{2} ⟩_{ρ_{ϵ}} + σ_{ϵ}^{2} .$

Proposition: Intrinsic and measured concentration are different

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