Manuscript Oai069-A@1 · Oai069-A.tex · section · a4913aab

The simultaneous certificate

Complete coverage and simultaneous finite-confidence certification › The simultaneous certificate · lines 1140–1180

Oai069-A@1#eq:intervals

For $0 < α_{s} < 1$ , define

t_{ν k P} r_{ν k P} a_{ν k P} = \frac{1}{N _{ν k P}} i \sum Z_{ν k P, i}, = \frac{2}{N _{ν k P}} lo g \frac{2 M}{α _{s}}, = r_{ν k P} + b_{ν k P} .

Set $t_{ν k I} = 1$ exactly and form

J_{ν, k} = \frac{1}{D} P \sum t_{ν k P} P .

It need not be a physical density matrix. No physicality projection is assumed.

Put

R_{ν, k} δ_{ν, k} L_{ν, k} U_{ν, k} = P \neq = I \sum a_{ν k P}^{2}^{1/2}, = \frac{1}{2} ∥ J_{ν, k} - J (T_{k}) ∥_{1}, = [δ_{ν, k} - \frac{1}{2} R_{ν, k} - b^{tr}_{ν, k}]_{[0, 1]}, = min {1, n (δ_{ν, k} + \frac{1}{2} R_{ν, k}) + b^{tr}_{ν, k}} .

Here $[x]_{[0, 1]} = max {0, min {1, x}}$ . For a complete direct-sum task set

L_{ν} = k max L_{ν, k}, U_{ν} = k max U_{ν, k} .

Displayed formulas in this passage

unlabeled · line 1143
$t_{ν k P} r_{ν k P} a_{ν k P} = \frac{1}{N _{ν k P}} i \sum Z_{ν k P, i}, = \frac{2}{N _{ν k P}} lo g \frac{2 M}{α _{s}}, = r_{ν k P} + b_{ν k P} .$
Oai069-A@1#eq:intervals · line 1160
$R_{ν, k} δ_{ν, k} L_{ν, k} U_{ν, k} = P \neq = I \sum a_{ν k P}^{2}^{1/2}, = \frac{1}{2} ∥ J_{ν, k} - J (T_{k}) ∥_{1}, = [δ_{ν, k} - \frac{1}{2} R_{ν, k} - b^{tr}_{ν, k}]_{[0, 1]}, = min {1, n (δ_{ν, k} + \frac{1}{2} R_{ν, k}) + b^{tr}_{ν, k}} .$
unlabeled · line 1152
$J_{ν, k} = \frac{1}{D} P \sum t_{ν k P} P .$
unlabeled · line 1176
$L_{ν} = k max L_{ν, k}, U_{ν} = k max U_{ν, k} .$