Manuscript Em03-B@2 · Em03-B-v2.tex · lemma · c1a0977d

Lemma: Gaussian mode-mismatch certificate

Finite switching and nonvacuum mode mismatch › Reference-stable mode comparison with a nonvacuum background · lines 898–941 · proof included or linked

Em03-B@2#lem:modeerror, Em03-B@2#eq:modeerror

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Suppose the $h, g$ marginal is centered, gauge-invariant, and quasifree, with occupations $E_{h}, E_{g}$ . For every normal spectator–reference extension,

d_{tr} (mode- f_{L} reduction, mode- h reduction) \leq μ_{L} := min {1, θ_{L} E_{h} + E_{g} + 6 E_{h} E_{g}} .

For the radiation marginal, $E_{h} = ν_{h}$ , and $μ_{L} \to 0$ . No independence between the two modes is required.

Proof. Purify the joint input, retaining its external systems. The unitary comparison and partial-trace contractivity give

d_{tr} \leq ∥ (V_{θ} - I) ∣ Ψ ⟩ ∥ \leq ∣ θ ∣ ⟨ J^{2} ⟩ .

The last inequality follows from the scalar spectral inequality $∣ e^{- i θ x} - 1∣ \leq ∣ θ x ∣$ .

Put $m = ⟨ a (g)^{†} a (h)⟩$ . Wick's rule gives

⟨ J^{2} ⟩ = E_{h} + E_{g} + 2 E_{h} E_{g} + 2∣ m ∣^{2} - 4ℜ (m^{2}) .

Cauchy–Schwarz gives $∣ m ∣^{2} \leq E_{h} E_{g}$ , hence $⟨ J^{2} ⟩ \leq E_{h} + E_{g} + 6 E_{h} E_{g}$ . All these moments depend only on the two-mode marginal, so the bound is unchanged by its reference extension.

For the radiation input,

θ_{L}^{2} (ν_{h} + E_{g} + 6 ν_{h} E_{g}) \to 0

by Lemma lem:window. This proves convergence.

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Em03-B@2#eq:modeerror · line 903
$d_{tr} (mode- f_{L} reduction, mode- h reduction) \leq μ_{L} := min {1, θ_{L} E_{h} + E_{g} + 6 E_{h} E_{g}} .$
unlabeled · line 918
$d_{tr} \leq ∥ (V_{θ} - I) ∣ Ψ ⟩ ∥ \leq ∣ θ ∣ ⟨ J^{2} ⟩ .$
unlabeled · line 927
$⟨ J^{2} ⟩ = E_{h} + E_{g} + 2 E_{h} E_{g} + 2∣ m ∣^{2} - 4ℜ (m^{2}) .$
unlabeled · line 937
$θ_{L}^{2} (ν_{h} + E_{g} + 6 ν_{h} E_{g}) \to 0$

Lemma: Gaussian mode-mismatch certificate

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