Manuscript X-Enter-ST@1 · X-Enter-ST.tex · proposition · f76acc1f

Proposition: Approximate orbit constancy

Stage B: the encoding-independent orbit obstruction › A quantitative approximate version · lines 535–572 · proof included or linked

X-Enter-ST@1#prop:approxorbit

Suppose the output is a normalized finite classical–quantum state $Ω (a)$ and

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

Then

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

for all real $a, b$ .

For any finite label family $a_{x}$ , prior $π_{x}$ , and estimator based on the declared output,

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

Proof. The first conclusion follows by the same substitution as in Theorem thm:orbit; the second follows by the triangle inequality.

Let $M_{x}$ be the estimator's POVM effects. Trace distance bounds an effect expectation, so

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

The first sum is at most $max_{x} π_{x} \sum_{x} Tr (M_{x} Ω (0)) = max_{x} π_{x}$ .

Displayed formulas in this passage

unlabeled · line 539
$λ \in R sup d_{tr} (Ω (λ c_{ε}), Ω (0)) \leq η .$
unlabeled · line 545
$d_{tr} (Ω (a), Ω (0)) \leq η, d_{tr} (Ω (a), Ω (b)) \leq 2 η$
unlabeled · line 553
$Pr {x = x} \leq min {1, x max π_{x} + η} .$
unlabeled · line 566
$x \sum π_{x} Tr (M_{x} Ω (a_{x})) \leq x \sum π_{x} Tr (M_{x} Ω (0)) + η .$