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

Lemma: Referenced multimode vacuum-loss comparison

A shared-port two-memory capture construction › Preparation, loss, and joint capture error · lines 1266–1289 · proof included or linked

Em03-C@1#lem:jointloss

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Vacuum pure loss of common transmissivity $t$ on a finite mode family satisfies

d_{tr} ((L_{t} \otimes id) (ρ), ρ) \leq ℓ (E, t) := min {1, 2 E (1 - t)}

when $E = tr (ρ N_{tot}) < \infty$ .

Proof. Purify the input and append vacuum loss modes. The overlap of the initial and beam-splitter-evolved purifications is

a = tr (ρ t^{N_{tot} /2}) .

For $0 \leq r \leq 1$ , $1 - r^{n} \leq n (1 - r)$ , so $1 - a \leq E (1 - t)$ . The pure-state distance is $1 - a^{2} \leq 2 (1 - a)$ . Partial trace proves the assertion.

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unlabeled · line 1270
$d_{tr} ((L_{t} \otimes id) (ρ), ρ) \leq ℓ (E, t) := min {1, 2 E (1 - t)}$
unlabeled · line 1281
$a = tr (ρ t^{N_{tot} /2}) .$

Lemma: Referenced multimode vacuum-loss comparison

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