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

Lemma: Coherent carrier approximation

C1 interface: joint carriers and referenced records › A different tail for coherent comparisons · lines 765–806 · proof included or linked

Em03-C@1#lem:coherenttail, Em03-C@1#eq:coherenttail

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For any normal packet–reference input,

d_{tr} (ρ_{12 S R}, (ι \otimes I_{S R}) (B_{a} \otimes id_{S R}) (ρ) (ι^{†} \otimes I_{S R})) \leq t (p_{a}),

where

t (p) = min {1, p + p /2} .

Proof. The embedded carrier output is

P_{a} ρ P_{a} + T, T ⪰ 0, tr T = p_{a} .

This follows by separating the low–low Kraus map from all other Kraus maps.

Purify $ρ$ to $∣ Ψ ⟩$ , and set $∣ Φ ⟩ = (P_{a} \otimes I) ∣ Ψ ⟩$ . Writing $p = p_{a}$ , one has $∥Φ ∥^{2} = ⟨ Ψ, Φ ⟩ = 1 - p$ . The rank-two calculation gives

∣ Ψ ⟩ ⟨ Ψ ∣ - ∣ Φ ⟩ ⟨ Φ ∣_{1} = 4 p - 3 p^{2} \leq 2 p .

Partial trace contracts this norm. Adding $T$ costs at most $p$ in trace norm. Divide by two and cap at one.

Displayed formulas in this passage

Em03-C@1#eq:coherenttail · line 768
$d_{tr} (ρ_{12 S R}, (ι \otimes I_{S R}) (B_{a} \otimes id_{S R}) (ρ) (ι^{†} \otimes I_{S R})) \leq t (p_{a}),$
unlabeled · line 781
$t (p) = min {1, p + p /2} .$
unlabeled · line 788
$P_{a} ρ P_{a} + T, T ⪰ 0, tr T = p_{a} .$
unlabeled · line 800
$∣ Ψ ⟩ ⟨ Ψ ∣ - ∣ Φ ⟩ ⟨ Φ ∣_{1} = 4 p - 3 p^{2} \leq 2 p .$

Lemma: Coherent carrier approximation

Displayed formulas in this passage

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