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

Lemma: Overlap with a pure target

Standard finite-dimensional conversions · lines 436–460 · proof included or linked

Oai265-B@2#lem:pure, Oai265-B@2#eq:pure

For a state $ρ$ , unit vector $ϕ$ , and $P = ∣ ϕ ⟩ ⟨ ϕ ∣$ , let $F = Tr (P ρ)$ . Then

1 - F \leq d_{tr} (ρ, P) \leq 1 - F .

Proof. The effect $P$ gives the lower bound. Write $ρ = \sum_{k} p_{k} ∣ ψ_{k} ⟩ ⟨ ψ_{k} ∣$ . Diagonalization on the span of $ϕ, ψ_{k}$ gives

d_{tr} (∣ ψ_{k} ⟩ ⟨ ψ_{k} ∣, P) = 1 - ∣ ⟨ ϕ, ψ_{k} ⟩ ∣^{2} .

Convexity of trace norm and concavity of square root imply

d_{tr} (ρ, P) \leq k \sum p_{k} 1 - ∣ ⟨ ϕ, ψ_{k} ⟩ ∣^{2} \leq 1 - F .

Displayed formulas in this passage

Oai265-B@2#eq:pure · line 440
$1 - F \leq d_{tr} (ρ, P) \leq 1 - F .$
unlabeled · line 450
$d_{tr} (∣ ψ_{k} ⟩ ⟨ ψ_{k} ∣, P) = 1 - ∣ ⟨ ϕ, ψ_{k} ⟩ ∣^{2} .$
unlabeled · line 455
$d_{tr} (ρ, P) \leq k \sum p_{k} 1 - ∣ ⟨ ϕ, ψ_{k} ⟩ ∣^{2} \leq 1 - F .$