Manuscript Hawk02-A@1 · Hawk02-A.tex · lemma · 7fbae745

Lemma: Flagged truncation of a commuting state

Declared objects, references, and cutoffs · lines 294–326 · proof included or linked

Hawk02-A@1#lem:flag

Let $P$ be an orthogonal projection, let $Q = I - P$ , and suppose $[ρ, P] = 0$ . Embed $ρ$ into $H \oplus C ∣ ⊥ ⟩$ and define

F_{P} (ρ) = P ρP + Tr (Qρ) ∣ ⊥ ⟩ ⟨ ⊥ ∣ .

Then $F_{P}$ is a CPTP map and

d_{tr} (ρ, F_{P} (ρ)) = Tr (Qρ) .

This equality need not hold for a noncommuting input.

Proof. Compression and the flagged trace term are completely positive, and their traces sum to $Tr ρ$ . For a commuting input, the difference has orthogonal blocks $QρQ$ and $- Tr (Qρ) ∣ ⊥ ⟩ ⟨ ⊥ ∣$ , each with trace norm $Tr (Qρ)$ .

For the last assertion take $∣ ψ ⟩ = 1 - p ∣ 0 ⟩ + p ∣ 1 ⟩$ , $0 < p < 1$ , and $P = ∣ 0 ⟩ ⟨ 0 ∣$ . Direct diagonalization gives

d_{tr} (∣ ψ ⟩ ⟨ ψ ∣, F_{P} (∣ ψ ⟩ ⟨ ψ ∣)) = \frac{4 p - 3 p ^{2} + p}{2} > p .

Displayed formulas in this passage

unlabeled · line 299
$F_{P} (ρ) = P ρP + Tr (Qρ) ∣ ⊥ ⟩ ⟨ ⊥ ∣ .$
unlabeled · line 305
$d_{tr} (ρ, F_{P} (ρ)) = Tr (Qρ) .$
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$d_{tr} (∣ ψ ⟩ ⟨ ψ ∣, F_{P} (∣ ψ ⟩ ⟨ ψ ∣)) = \frac{4 p - 3 p ^{2} + p}{2} > p .$

Lemma: Flagged truncation of a commuting state

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