Manuscript Em03-B@2 · Em03-B-v2.tex · section · c1a0977d

Binary register and a total decoder

The B1–B3 mathematical interface › Binary register and a total decoder · lines 524–573

Em03-B@2#eq:idealread, Em03-B@2#eq:decoder

Set

b_{N} = ⌈ lo g_{2} (N + 1) ⌉, d_{N} = 2^{b_{N}}, Q_{N} = (C^{2})^{\otimes b_{N}} .

Identify carrier basis $k$ with binary integer $k$ in $Q$ ; write the isometry as $J_{N}$ , and set

B_{N} (X) = J_{N} C_{N} (X) J_{N}^{†} .

For every $0 \leq j < d_{N}$ , including unused words, define

c_{N} (j) = min (j, N), y_{j} = (bin_{b_{N}} (j), c_{N} (j)) .

Let $Y$ contain all distinct $y_{j}$ and any declared failure symbols. The full record space is $H_{F} = C^{Y}$ . Ideal readout has zero weight on failures.

For a spectator $S$ , define

M_{j} = ⟨ j ∣_{Q} \otimes I_{S}, E_{j} = M_{j}^{†} M_{j} .

The selective map is $X \mapsto M_{j} X M_{j}^{†}$ , whereas its effect is $E_{j}$ . Define

P_{N}^{S} (X) = j = 0 \sum^{d_{N} - 1} ∣ y_{j} ⟩ ⟨ y_{j} ∣_{F} \otimes M_{j} X M_{j}^{†} .

Assign every failure symbol a fixed decoded value in ${0, \dots, N}$ ; below all failure values are zero. Write this total decoder as $d (y)$ , with $d (y_{j}) = c_{N} (j)$ . Its channel on arbitrary record operators is

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

This is CPTP on the full record space, not merely a prescription on its populated diagonal.

Displayed formulas in this passage

Em03-B@2#eq:idealread · line 554
$P_{N}^{S} (X) = j = 0 \sum^{d_{N} - 1} ∣ y_{j} ⟩ ⟨ y_{j} ∣_{F} \otimes M_{j} X M_{j}^{†} .$
Em03-B@2#eq:decoder · line 565
$G_{N} (X) = y \in Y \sum ⟨ y ∣ X ∣ y ⟩ ∣ d (y)⟩ ⟨ d (y) ∣_{rec} .$
unlabeled · line 527
$b_{N} = ⌈ lo g_{2} (N + 1) ⌉, d_{N} = 2^{b_{N}}, Q_{N} = (C^{2})^{\otimes b_{N}} .$
unlabeled · line 533
$B_{N} (X) = J_{N} C_{N} (X) J_{N}^{†} .$
unlabeled · line 538
$c_{N} (j) = min (j, N), y_{j} = (bin_{b_{N}} (j), c_{N} (j)) .$
unlabeled · line 548
$M_{j} = ⟨ j ∣_{Q} \otimes I_{S}, E_{j} = M_{j}^{†} M_{j} .$