Manuscript Oai069-A@1 · Oai069-A.tex · section · a4913aab

A complete decoder using only the receiver

Phase 2: positive realization and access-restricted recovery › A complete decoder using only the receiver · lines 700–727

Oai069-A@1#eq:decoder

Write a received operator, after $F_{d}$ , as $Y$ . Define

D_{12} (Y) = k \sum ∣ k ⟩ ⟨ k ∣_{J_{out}} \otimes Tr_{R_{2}} [U_{k}^{†} (⟨ k ∣_{J} Y ∣ k ⟩_{J}) U_{k}] .

This is defined on the entire receiver matrix algebra, not only on the code range.

The programme is:

apply the printed $F_{d}$ to $S$ , unless character coordinates have already been supplied;
read $J$ in the computational basis, retaining every outcome $k$ ;
on that branch apply $CNOT_{1 \to 2}$ , then $H$ on $R_{1}$ , then $Z^{ϵ_{k}}$ on $R_{1}$ ;
discard $R_{2}$ , return $R_{1}$ as $C$ , and return $k$ as the classical label.

The order in step three is $U_{k}^{†}$ .

Displayed formulas in this passage

Oai069-A@1#eq:decoder · line 703
$D_{12} (Y) = k \sum ∣ k ⟩ ⟨ k ∣_{J_{out}} \otimes Tr_{R_{2}} [U_{k}^{†} (⟨ k ∣_{J} Y ∣ k ⟩_{J}) U_{k}] .$