Manuscript En10-A@1 · En10-A.tex · section · 0a1d7743

The finite Enter theorem and estimation contract

The finite Enter theorem and estimation contract · lines 575–615

En10-A@1#eq:statistic, En10-A@1#eq:estimator, En10-A@1#eq:decoder-channel

For the complete word $r_{i} = (x_{i}, y_{i})$ , define

Z_{i} = 1_{{y_{i} = 0}}, K = i = 1 \sum^{n} Z_{i}, Z = \frac{K}{n} .

The separate classical inference map returns

θ = Π_{[0, 1]} (\frac{Z - a}{b}), Π_{[0, 1]} (u) = max {0, min {1, u}} .

For definiteness, this is a classical channel from word lists to integer labels $k \in {0, \dots, n}$ , with the declared interpretation

k ⟼ θ_{k} = Π_{[0, 1]} (\frac{k / n - a}{b}) .

A CPTP matrix extension is

D_{n} (X) = r \in R^{n} \sum ⟨ r ∣ X ∣ r ⟩ ∣ K (r)⟩ ⟨ K (r)∣ .

Its Kraus operators are $∣ K (r)⟩ ⟨ r ∣$ , whose squared moduli sum to the identity. This map does not return a data qubit.

Displayed formulas in this passage

En10-A@1#eq:statistic · line 578
$Z_{i} = 1_{{y_{i} = 0}}, K = i = 1 \sum^{n} Z_{i}, Z = \frac{K}{n} .$
En10-A@1#eq:estimator · line 585
$θ = Π_{[0, 1]} (\frac{Z - a}{b}), Π_{[0, 1]} (u) = max {0, min {1, u}} .$
En10-A@1#eq:decoder-channel · line 604
$D_{n} (X) = r \in R^{n} \sum ⟨ r ∣ X ∣ r ⟩ ∣ K (r)⟩ ⟨ K (r)∣ .$
unlabeled · line 597
$k ⟼ θ_{k} = Π_{[0, 1]} (\frac{k / n - a}{b}) .$