Manuscript Em03-C@1 · Em03-C-v1.tex · section · 47dd71ec

Complete raw alphabet and decoding

C1 interface: joint carriers and referenced records › Complete raw alphabet and decoding · lines 644–681

Em03-C@1#eq:totaldecoder

For raw integers $j_{i} \in {0, \dots, d_{i} - 1}$ , put

c_{i} (j_{i}) = min (j_{i}, a_{i}) .

An ordinary record contains both raw words, both decoded integers, and an “ordinary” status:

y_{j} = (bin (j_{1}), bin (j_{2}), c_{1} (j_{1}), c_{2} (j_{2}), ordinary) .

Enlarge this alphabet $Y$ by all declared failure records. Failure records retain their labels; their decoded pair is $(0, 0)$ . A different bounded assignment could be specified, but must be fixed before using a mean certificate.

The record can be embedded into $F_{1} F_{2} F_{flag}$ . We denote the whole classical output by $F$ . For arbitrary record operators, the total decoder is

G_{a} (X) = y \in Y \sum ⟨ y ∣ X ∣ y ⟩ ∣ c (y)⟩ ⟨ c (y) ∣_{rec} .

It is CPTP on the full alphabet.

For computational destructive readout, set

M_{j} = ⟨ j_{1}, j_{2} ∣_{Q} \otimes I_{S}, P_{Z}^{S} (X) = j \sum ∣ y_{j} ⟩ ⟨ y_{j} ∣ \otimes M_{j} X M_{j}^{†} .

The measured register is absent from the output.

Displayed formulas in this passage

Em03-C@1#eq:totaldecoder · line 665
$G_{a} (X) = y \in Y \sum ⟨ y ∣ X ∣ y ⟩ ∣ c (y)⟩ ⟨ c (y) ∣_{rec} .$
unlabeled · line 647
$c_{i} (j_{i}) = min (j_{i}, a_{i}) .$
unlabeled · line 652
$y_{j} = (bin (j_{1}), bin (j_{2}), c_{1} (j_{1}), c_{2} (j_{2}), ordinary) .$
unlabeled · line 674
$M_{j} = ⟨ j_{1}, j_{2} ∣_{Q} \otimes I_{S}, P_{Z}^{S} (X) = j \sum ∣ y_{j} ⟩ ⟨ y_{j} ∣ \otimes M_{j} X M_{j}^{†} .$