Manuscript Hawk04-B@2 · Hawk04-B.tex · proposition · ba3b0713

Proposition: Commuting couplings give measure-and-prepare radiation

The commuting four-Majorana control › The general commuting obstruction · lines 1792–1821 · proof included or linked

Hawk04-B@2#prop:QND

Suppose

[H_{S}, A_{a}] = 0, [A_{a}, A_{b}] = 0

for every pair. For any isometric diary insertion and fixed bath preparation, the collected-reservoir channel is measure-and-prepare. Every deterministic radiation decoder has error at least $1 - d^{- 1}$ .

Proof. Choose simultaneous source eigenspace projectors $P_{λ}$ . The evolution has blocks

U (t) = λ \sum P_{λ} \otimes U_{λ} (t) .

After tracing the source, cross-projector terms vanish. Including the retained carrier in the trace gives

N_{t} (ρ) = λ \sum Tr (F_{λ} ρ) σ_{λ, t}, F_{λ} = W^{†} (P_{λ} \otimes I_{D}) W .

The effects are positive and sum to $I_{C}$ . Theorem thm:classical applies.

Displayed formulas in this passage

unlabeled · line 1795
$[H_{S}, A_{a}] = 0, [A_{a}, A_{b}] = 0$
unlabeled · line 1807
$U (t) = λ \sum P_{λ} \otimes U_{λ} (t) .$
unlabeled · line 1812
$N_{t} (ρ) = λ \sum Tr (F_{λ} ρ) σ_{λ, t}, F_{λ} = W^{†} (P_{λ} \otimes I_{D}) W .$

Nearby in this section

section: The general commuting obstruction