Manuscript Hawk04-D@1 · Hawk04-D-v1.tex · definition · 39b16f67

Definition: Complete interface and diagnostics

Complete interfaces and the finite operational core › Definitions · lines 290–325

Hawk04-D@1#def:interface

A complete interface is an isometry

V : C ⟶ R \otimes B, V^{†} V = I_{C} .

Its complete, accessible, and complementary channels are

V (X) = V X V^{†}, N (X) = Tr_{B} V (X), E (X) = Tr_{R} V (X) .

The complete inaccessible output $B$ includes all unreturned preparation purifiers, source registers, uncollected modes, apparatus outputs, work registers, and discarded-record stores in the declared model. The independent reference $Q$ is not part of $B$ .

For a state $σ$ , write

T_{σ} (X) = Tr (X) σ .

Use half-normalized distances

d_{tr} (ρ, η) = \frac{1}{2} ∥ ρ - η ∥_{1}, d_{⋄} (Φ, Ψ) = \frac{1}{2} ∥ Φ - Ψ ∥_{⋄},

and define

ε_{rec} (N) δ_{full} (E) = D : R \to C CPTP in f d_{⋄} (DN, id_{C}), = σ \in D (B) in f d_{⋄} (E, T_{σ}) .

Decoders are deterministic and return a quantum code. A normalized successful branch alone is not a decoder.

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unlabeled · line 317
$ε_{rec} (N) δ_{full} (E) = D : R \to C CPTP in f d_{⋄} (DN, id_{C}), = σ \in D (B) in f d_{⋄} (E, T_{σ}) .$
unlabeled · line 293
$V : C ⟶ R \otimes B, V^{†} V = I_{C} .$
unlabeled · line 297
$V (X) = V X V^{†}, N (X) = Tr_{B} V (X), E (X) = Tr_{R} V (X) .$
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$T_{σ} (X) = Tr (X) σ .$
unlabeled · line 312
$d_{tr} (ρ, η) = \frac{1}{2} ∥ ρ - η ∥_{1}, d_{⋄} (Φ, Ψ) = \frac{1}{2} ∥ Φ - Ψ ∥_{⋄},$

Definition: Complete interface and diagnostics

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