Manuscript Em03-G@1 · Em03-G-v1.tex · lemma · 0bb3a8a3

Lemma: A product-reset comparison

A phase-matched packet model and an explicit decoder · lines 1176–1203 · proof included or linked

Em03-G@1#lem:reset

For a state $ζ_{S B}$ , let $P = ∣ χ ⟩ ⟨ χ ∣ \otimes I$ , and $p = 1 - tr (P ζ)$ . Then

d_{tr} (ζ, ∣ χ ⟩ ⟨ χ ∣ \otimes ζ_{B}) \leq t (p) := min {1, p + p /2} .

The bound retains arbitrary systems in $B$ .

Proof. Purify $ζ$ and apply $P$ . The rank-two pure-state calculation, followed by partial trace, gives

\frac{1}{2} ∥ ζ - P ζ P ∥_{1} \leq p .

Moreover

∣ χ ⟩ ⟨ χ ∣ \otimes ζ_{B} = P ζ P + ∣ χ ⟩ ⟨ χ ∣ \otimes ζ_{B}^{bad},

where $ζ_{B}^{bad} = Tr_{S} ((I - P) ζ (I - P)) ⪰ 0$ has trace $p$ . This additional term costs $p /2$ .

Displayed formulas in this passage

unlabeled · line 1181
$d_{tr} (ζ, ∣ χ ⟩ ⟨ χ ∣ \otimes ζ_{B}) \leq t (p) := min {1, p + p /2} .$
unlabeled · line 1191
$\frac{1}{2} ∥ ζ - P ζ P ∥_{1} \leq p .$
unlabeled · line 1195
$∣ χ ⟩ ⟨ χ ∣ \otimes ζ_{B} = P ζ P + ∣ χ ⟩ ⟨ χ ∣ \otimes ζ_{B}^{bad},$