Manuscript Em03-C@1 · Em03-C-v1.tex · lemma · 47dd71ec

Lemma: Joint Gaussian mode-selection error

Joint temporal modes and participating radiation › Principal-plane comparison · lines 1015–1066 · proof included or linked

Em03-C@1#lem:mode, Em03-C@1#eq:jointmode

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For the inherited Gaussian marginal on $v_{L}$ , and any spectator–reference extension,

d_{tr} (S_{F} (ρ), I_{H} (ρ)) \leq μ_{L} := min {1, r : θ_{r} > 0 \sum θ_{r} E_{h^{'}_{r}} + E_{g_{r}} + 6 E_{h^{'}_{r}} E_{g_{r}}} .

Here each occupation is computed from the actual port covariance. Moreover, $μ_{L} \to 0$ .

Proof. The rotation generator in plane $r$ is

J_{r} = i (a (g_{r})^{†} a (h^{'}_{r}) - a (h^{'}_{r})^{†} a (g_{r})) .

These generators are self-adjoint on their total-number-sector direct sums. Purification, the spectral inequality $∣ e^{i x} - 1∣ \leq ∣ x ∣$ , and the vector triangle inequality give the bound

d_{tr} \leq r \sum θ_{r} ⟨ J_{r}^{2} ⟩ .

For a centered gauge-invariant Gaussian two-mode marginal, put $m = ⟨ a (g_{r})^{†} a (h^{'}_{r})⟩$ . Wick contraction gives

⟨ J_{r}^{2} ⟩ = E_{h^{'}_{r}} + E_{g_{r}} + 2 E_{h^{'}_{r}} E_{g_{r}} + 2∣ m ∣^{2} - 4 Re (m^{2}) \leq E_{h^{'}_{r}} + E_{g_{r}} + 6 E_{h^{'}_{r}} E_{g_{r}},

using $∣ m ∣^{2} \leq E_{h^{'}_{r}} E_{g_{r}}$ . This depends only on the participating marginal, so arbitrary reference extensions obey the same estimate.

The principal target occupations are bounded by $n_{m a x}$ . The residual vectors $f^{'}_{r} - cos θ_{r} h^{'}_{r}$ have weighted occupation tending uniformly to zero by Lemma lem:windows. Their occupations are $sin^{2} θ_{r} E_{g_{r}}$ . Therefore $θ_{r}^{2} E_{g_{r}} \to 0$ , proving the last assertion.

Displayed formulas in this passage

Em03-C@1#eq:jointmode · line 1019
$d_{tr} (S_{F} (ρ), I_{H} (ρ)) \leq μ_{L} := min {1, r : θ_{r} > 0 \sum θ_{r} E_{h^{'}_{r}} + E_{g_{r}} + 6 E_{h^{'}_{r}} E_{g_{r}}} .$
unlabeled · line 1035
$J_{r} = i (a (g_{r})^{†} a (h^{'}_{r}) - a (h^{'}_{r})^{†} a (g_{r})) .$
unlabeled · line 1043
$d_{tr} \leq r \sum θ_{r} ⟨ J_{r}^{2} ⟩ .$
unlabeled · line 1050
$⟨ J_{r}^{2} ⟩ = E_{h^{'}_{r}} + E_{g_{r}} + 2 E_{h^{'}_{r}} E_{g_{r}} + 2∣ m ∣^{2} - 4 Re (m^{2}) \leq E_{h^{'}_{r}} + E_{g_{r}} + 6 E_{h^{'}_{r}} E_{g_{r}},$

Lemma: Joint Gaussian mode-selection error

Displayed formulas in this passage

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