Manuscript Psy02-A@1 · Psy02-A.tex · section · d92ce402

Preparation in a moment-compatible metric

Input, preparation, and measurement budgets › Preparation in a moment-compatible metric · lines 1356–1377

Psy02-A@1#eq:prep-tail

For completeness, a trace-distance allowance becomes usable when supplemented by a tail certificate. Let

A_{L} = min {(B - o 1)^{2}, L^{2} 1}, T_{σ} (L) = Tr (σ [(B - o 1)^{2} - A_{L}]) .

If $d_{tr} (ρ, σ) \leq δ$ , then

Tr (σ (B - o 1)^{2}) \leq Tr (ρ (B - o 1)^{2}) + L^{2} δ + T_{σ} (L) .

Indeed $0 \leq A_{L} \leq L^{2} 1$ , and an effect expectation changes by at most trace distance.

Displayed formulas in this passage

Psy02-A@1#eq:prep-tail · line 1369
$Tr (σ (B - o 1)^{2}) \leq Tr (ρ (B - o 1)^{2}) + L^{2} δ + T_{σ} (L) .$
unlabeled · line 1361
$A_{L} = min {(B - o 1)^{2}, L^{2} 1}, T_{σ} (L) = Tr (σ [(B - o 1)^{2} - A_{L}]) .$