Manuscript Hawk02-E@1 · Hawk02-E.tex · proposition · 4841592b

Proposition: A specified decoder and complete extension

Complete-residual certificates and decoder certificates › Finite semidefinite certificates · lines 1350–1391 · proof included or linked

Hawk02-E@1#prop:decoder

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Let $D$ be a rigorously CPTP decoder on the finite packet output space, and define

r (t) = \frac{1}{2} J_{n} (D N_{t}) - ∣ Φ_{d} ⟩ ⟨ Φ_{d} ∣_{1} .

Complete $D$ to the full packet Fock space by

D (Y) = D (P_{R} Y P_{R}) + Tr ((I - P_{R}) Y) \frac{I _{C}}{d},

where $P_{R}$ projects onto the finite packet occupation space. Then

d_{⋄} (D N_{t}, id_{C}) \leq min {1, d r (t) + b_{reg}} .

Proof. Compression by $P_{R}$ is completely positive. The second term is a positive-effect measurement followed by state preparation. Their traces add to $Tr Y$ , so the extension is CPTP.

The approximate channel is supported on $P_{R}$ . Lemma lem:choi gives its decoded error $d r (t)$ . Postprocessing contraction and Theorem thm:regulator give the additional $b_{reg}$ .

Displayed formulas in this passage

unlabeled · line 1354
$r (t) = \frac{1}{2} J_{n} (D N_{t}) - ∣ Φ_{d} ⟩ ⟨ Φ_{d} ∣_{1} .$
unlabeled · line 1364
$D (Y) = D (P_{R} Y P_{R}) + Tr ((I - P_{R}) Y) \frac{I _{C}}{d},$
unlabeled · line 1373
$d_{⋄} (D N_{t}, id_{C}) \leq min {1, d r (t) + b_{reg}} .$

Proposition: A specified decoder and complete extension

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