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

Lemma: Referenced pure-loss bound

B4 capture: an explicit passive input–output interaction › A direct pure-loss benchmark · lines 1093–1118 · proof included or linked

Em03-B@2#lem:loss, Em03-B@2#eq:lossbound

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Let $L_{t}$ be the vacuum pure-loss channel of transmissivity $0 \leq t \leq 1$ . If $tr (ρN) \leq E$ , then, including any normal reference,

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

Proof. Purify $ρ$ , append a vacuum loss oscillator, and apply the beam splitter. The overlap with the initial purified state is

A = tr (ρ t^{N /2}) = n \sum p (n) (t)^{n} .

This is nonnegative. The inequality $1 - s^{n} \leq n (1 - s)$ , $0 \leq s \leq 1$ , gives $1 - A \leq E (1 - t)$ . The pure-state trace distance is $1 - A^{2} \leq 2 (1 - A)$ . Partial trace proves (eq:lossbound).

Displayed formulas in this passage

Em03-B@2#eq:lossbound · line 1098
$d_{tr} ((L_{t} \otimes id) (ρ), ρ) \leq ℓ (E, t) := min {1, 2 E (1 - t)} .$
unlabeled · line 1109
$A = tr (ρ t^{N /2}) = n \sum p (n) (t)^{n} .$

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section: A direct pure-loss benchmark