Manuscript Em03-E@1 · Em03-E-v1.tex · section · c4b07e6a

A complete decoder

E4: explicit decoder and optimal referenced recovery › A complete decoder · lines 791–820

On the physical payload support, define

D (Y) = a, b \sum P_{ab}^{†} J_{o}^{†} ⟨ a, b ∣ Tr_{O_{1} O_{2}} (Y) ∣ a, b ⟩ J_{o} P_{ab} + tr (P_{bad} Y) \frac{I _{2}}{2} .

Here

P_{good} = I_{O_{1} O_{2}} \otimes a, b \sum ∣ a, b ⟩ ⟨ a, b ∣_{C_{1}} \otimes J_{o} J_{o}^{†}, P_{bad} = I - P_{good} .

The first term has trace $tr (P_{good} Y)$ . Its Kraus maps are the count- and key-conditioned extraction maps. The second term is a replacement CP map of trace $tr (P_{bad} Y)$ . Hence $D$ is CPTP on every physical input, not only on emitted states.

The Pauli correction is applied before destructive logical readout. Discarding the count and key metadata does not discard the returned message.

Displayed formulas in this passage

unlabeled · line 794
$D (Y) = a, b \sum P_{ab}^{†} J_{o}^{†} ⟨ a, b ∣ Tr_{O_{1} O_{2}} (Y) ∣ a, b ⟩ J_{o} P_{ab} + tr (P_{bad} Y) \frac{I _{2}}{2} .$
unlabeled · line 803
$P_{good} = I_{O_{1} O_{2}} \otimes a, b \sum ∣ a, b ⟩ ⟨ a, b ∣_{C_{1}} \otimes J_{o} J_{o}^{†}, P_{bad} = I - P_{good} .$

Nearby in this section

proposition: Proposition: Decoded channel