Manuscript X-A-conv-ST@1 · X-A-conv-ST.tex · proposition · 8ca47088

Proposition: Approximate orbit bound

Stage C: readout closure of the retained coefficient › The encoding-independent statement · lines 1461–1493 · proof included or linked

X-A-conv-ST@1#prop:approx

Let $Ω (a)$ be normalized finite classical–quantum output states. If

λ sup d_{tr} (Ω (λ c_{ε}), Ω (0)) \leq η,

then, for all $a, b$ ,

d_{tr} (Ω (a), Ω (0)) \leq η, d_{tr} (Ω (a), Ω (b)) \leq 2 η .

For any finite hypothesis family $a_{x}$ , prior $π_{x}$ , and estimator using only this output,

Pr (x = x) \leq min {1, x max π_{x} + η} .

Proof. The first assertion uses the same substitution as Theorem thm:orbitclosure; the second uses the triangle inequality.

If $M_{x}$ are the estimator effects, trace distance bounds each effect expectation. Hence the success probability is at most

x \sum π_{x} Tr (M_{x} Ω (0)) + η \leq x max π_{x} + η .

Displayed formulas in this passage

unlabeled · line 1465
$λ sup d_{tr} (Ω (λ c_{ε}), Ω (0)) \leq η,$
unlabeled · line 1470
$d_{tr} (Ω (a), Ω (0)) \leq η, d_{tr} (Ω (a), Ω (b)) \leq 2 η .$
unlabeled · line 1476
$Pr (x = x) \leq min {1, x max π_{x} + η} .$
unlabeled · line 1489
$x \sum π_{x} Tr (M_{x} Ω (0)) + η \leq x max π_{x} + η .$