Manuscript Oai265-B@2 · Oai265-B-v2.tex · proposition · 94b6f42f

Proposition: Bell-overlap recovery bounds

Standard finite-dimensional conversions · lines 465–486 · proof included or linked

Oai265-B@2#prop:bellbounds, Oai265-B@2#eq:bellbounds, Oai265-B@2#eq:transferbounds

For a $d$ -dimensional logical channel, put $F_{e} = Tr (P_{d} J_{E})$ . Then

1 - F_{e} \leq e (E) \leq min {1, d 1 - F_{e}} .

Under (eq:transfer),

max {0, 1 - F_{e} - b} \leq e (E_{⋆}) \leq min {1, d 1 - F_{e} + b} .

Proof. Combine Lemmas lem:choi and lem:pure. The reverse triangle inequality gives $∣ e (E_{⋆}) - e (E) ∣ \leq b$ .

Displayed formulas in this passage

Oai265-B@2#eq:bellbounds · line 469
$1 - F_{e} \leq e (E) \leq min {1, d 1 - F_{e}} .$
Oai265-B@2#eq:transferbounds · line 474
$max {0, 1 - F_{e} - b} \leq e (E_{⋆}) \leq min {1, d 1 - F_{e} + b} .$