Manuscript En09-A@1 · En09-A.tex · section · eafec9de

Reference-stable comparison

Complete-channel stability and refinement independence › Reference-stable comparison · lines 1268–1291

En09-A@1#eq:zero-distance

For compatible finite linear maps define

∥ Λ ∥_{⋄} = r \geq 1 sup X \neq = 0 sup \frac{∥ ( Λ \otimes id _{r} ) ( X ) ∥ _{1}}{∥ X ∥ _{1}}, d_{⋄} (Φ, Ψ) = \frac{1}{2} ∥ Φ - Ψ ∥_{⋄} .

For states, $d_{tr} (ρ, σ) = \frac{1}{2} ∥ ρ - σ ∥_{1}$ . The half-diamond normalization gives $0 \leq d_{⋄} (Φ, Ψ) \leq 1$ for channels and includes arbitrary reference-entangled inputs Watrous (Chapter 3).

By (eq:exact-channel-identity),

d_{⋄} (M_{C}, J_{C}) = 0.

This is equality on the full fixed interface, not merely agreement at one calibrated preparation.

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En09-A@1#eq:zero-distance · line 1285
$d_{⋄} (M_{C}, J_{C}) = 0.$
unlabeled · line 1271
$∥ Λ ∥_{⋄} = r \geq 1 sup X \neq = 0 sup \frac{∥ ( Λ \otimes id _{r} ) ( X ) ∥ _{1}}{∥ X ∥ _{1}}, d_{⋄} (Φ, Ψ) = \frac{1}{2} ∥ Φ - Ψ ∥_{⋄} .$

Reference-stable comparison

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