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

Proposition: Robust estimation under complete-channel error

Implementation tolerance on the same interface › A finite-shot allowance · lines 1192–1239 · proof included or linked

En10-A@1#prop:robust, En10-A@1#eq:robust

Keep the nominal coefficients $a, b$ , source family, and fixed sample size. Suppose shot $i$ uses a classical-output CPTP map $I_{i}$ on the same comparison interface and

d_{⋄} (I_{i}, I) \leq δ (i = 1, \dots, n) .

Assume the resulting shot records are independent; they need not be identically distributed. Apply the nominal inference map (eq:estimator). Then

θ Pr {∣ θ - θ ∣ \leq min {1, \frac{δ + lo g ( 2/ α ) / ( 2 n )}{b}}} \geq 1 - α .

Proof. Let $q_{i} = E_{θ} Z_{i}$ for the implemented shot. Trace-distance control of the output states bounds every event probability difference, so

∣ q_{i} - q_{θ} ∣ \leq δ .

Consequently

\frac{1}{n} i \sum q_{i} - q_{θ} \leq δ .

The bounded-independent-variable argument in the main proof applies without identical means and gives

Pr {Z - \frac{1}{n} i \sum q_{i} > \frac{lo g ( 2/ α )}{2 n}} \leq α .

Combine these inequalities and apply the same projection argument as before.

Displayed formulas in this passage

En10-A@1#eq:robust · line 1204
$θ Pr {∣ θ - θ ∣ \leq min {1, \frac{δ + lo g ( 2/ α ) / ( 2 n )}{b}}} \geq 1 - α .$
unlabeled · line 1197
$d_{⋄} (I_{i}, I) \leq δ (i = 1, \dots, n) .$
unlabeled · line 1220
$∣ q_{i} - q_{θ} ∣ \leq δ .$
unlabeled · line 1224
$\frac{1}{n} i \sum q_{i} - q_{θ} \leq δ .$
unlabeled · line 1229
$Pr {Z - \frac{1}{n} i \sum q_{i} > \frac{lo g ( 2/ α )}{2 n}} \leq α .$

Nearby in this section

section: A finite-shot allowance