Manuscript Oai265-B@2 · Oai265-B-v2.tex · proposition · 94b6f42f

Proposition: Computable confidence radius

A tighter channel-confidence-region route › Explicit qubit Choi confidence region · lines 1522–1549 · proof included or linked

Oai265-B@2#prop:radius, Oai265-B@2#eq:radius

On $J_{E} \in C$ , choose any feasible $J \in C$ , and let

t_{k} = ω_{k} + ∣ Tr (P_{k} J) - a_{k} ∣.

Then the corresponding channel obeys

d_{⋄} (E, E) \leq η, η = min ⎩ ⎨ ⎧ 1, k = 1 \sum^{15} t_{k}^{2} ⎭ ⎬ ⎫, t_{k} \leq 2 ω_{k} .

The feasible center may depend on the data.

Proof. For $H = J_{E} - J$ , write

H = \frac{1}{4} k \sum h_{k} P_{k}, ∣ h_{k} ∣ \leq t_{k}, ∥ H ∥_{2} = \frac{1}{2} k \sum h_{k}^{2} .

In dimension four, $∥ H ∥_{1} \leq 2 ∥ H ∥_{2}$ . Lemma lem:choi with $d = 2$ gives $d_{⋄} (E, E) \leq ∥ H ∥_{1}$ .

Displayed formulas in this passage

Oai265-B@2#eq:radius · line 1530
$d_{⋄} (E, E) \leq η, η = min ⎩ ⎨ ⎧ 1, k = 1 \sum^{15} t_{k}^{2} ⎭ ⎬ ⎫, t_{k} \leq 2 ω_{k} .$
unlabeled · line 1526
$t_{k} = ω_{k} + ∣ Tr (P_{k} J) - a_{k} ∣.$
unlabeled · line 1542
$H = \frac{1}{4} k \sum h_{k} P_{k}, ∣ h_{k} ∣ \leq t_{k}, ∥ H ∥_{2} = \frac{1}{2} k \sum h_{k}^{2} .$