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

Proposition: Coherent marginal comparison

Quantitative stability of the classical inputs › Sharper coherent spectral stability · lines 1283–1306 · proof included or linked

Em06-B@1#prop:Gaussian-stability, Em06-B@1#eq:Gaussian-stability

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For two displaced-vacuum preparations with directions $v_{0}, v_{1}$ and means $o_{0}, o_{1}$ , their marginal laws satisfy

W_{2} (μ_{0, ϵ}, μ_{1, ϵ})^{2} W_{2} (μ_{0, ϵ}, μ_{1, ϵ}) = ∣ o_{1} - o_{0} ∣^{2} + ϵ (∥ v_{1} ∥ - ∥ v_{0} ∥)^{2}, \leq ∣ o_{1} - o_{0} ∣ + ϵ ∥ v_{1} - v_{0} ∥.

Proof. The marginal laws are $N (o_{a}, ϵ ∥ v_{a} ∥^{2})$ . Couple them using the same real standard Gaussian. This gives the first displayed quantity as an upper bound. For any coupling, Cauchy–Schwarz bounds its covariance by the product of its standard deviations, giving the matching lower bound. The reverse triangle inequality for Hilbert norms proves the second statement.

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Em06-B@1#eq:Gaussian-stability · line 1287
$W_{2} (μ_{0, ϵ}, μ_{1, ϵ})^{2} W_{2} (μ_{0, ϵ}, μ_{1, ϵ}) = ∣ o_{1} - o_{0} ∣^{2} + ϵ (∥ v_{1} ∥ - ∥ v_{0} ∥)^{2}, \leq ∣ o_{1} - o_{0} ∣ + ϵ ∥ v_{1} - v_{0} ∥.$

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section: Sharper coherent spectral stability