Manuscript XVI-O1-ST@5 · XVI-O1-ST-v5.tex · proposition · dd45e0d1

Proposition: Complete-channel implementation budget

Numerical reports and sharp complete error · lines 686–702 · proof included or linked

XVI-O1-ST@5#prop:budget

For compatible ideal and implemented complete channels,

d_{⋄} (Φ_{n} \dots Φ_{1}, Φ_{n} \dots Φ_{1}) \leq min {1, j = 1 \sum^{n} d_{⋄} (Φ_{j}, Φ_{j})} .

Proof. Telescope the difference, use contractivity under channel composition and the triangle inequality, and use the maximum one for half-diamond channel distance Watrous.

Displayed formulas in this passage

unlabeled · line 689
$d_{⋄} (Φ_{n} \dots Φ_{1}, Φ_{n} \dots Φ_{1}) \leq min {1, j = 1 \sum^{n} d_{⋄} (Φ_{j}, Φ_{j})} .$

Nearby in this section

section: Numerical reports and sharp complete error
theorem: Theorem: Sharp reference-stable numerical-report error
remark: Remark: Different report interfaces have different metrics
corollary: Corollary: Sparse errors are not diluted
section: Numerical reports and sharp complete error