Manuscript En04-A@1 · En04-A.tex · section · bab77211

Complete channels and residual response

Complete channels and residual response · lines 279–323

En04-A@1#eq:recovery, En04-A@1#eq:leakage

Let $C ≃ C^{d}$ , $d \geq 2$ , be a finite logical code. Output Hilbert spaces are separable. Channels act normally on trace-class operators and are completely positive and trace preserving, abbreviated CPTP.

For an isometry

V : C ⟶ R \otimes B,

set

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

The residual $B$ includes every inaccessible output: preparation purifiers, retained memories, uncollected modes, displaced carriers, and inaccessible apparatus factors.

Distances are

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

All channel statements retain arbitrary untouched references. For finite input, standard diamond-norm and Choi conventions are as in Watrous.

Define the replacement channel

T_{σ} (X) = Tr (X) σ

and the quantities

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

Displayed formulas in this passage

En04-A@1#eq:recovery · line 314
$ε_{rec}^{R} (V) δ_{full} (V) = D : R \to C CPTP in f d_{⋄} (DN, id_{C}), = σ in f d_{⋄} (C, T_{σ}) .$
En04-A@1#eq:leakage · line 314
$ε_{rec}^{R} (V) δ_{full} (V) = D : R \to C CPTP in f d_{⋄} (DN, id_{C}), = σ in f d_{⋄} (C, T_{σ}) .$
unlabeled · line 287
$V : C ⟶ R \otimes B,$
unlabeled · line 291
$V (X) = V X V^{†}, N (X) = Tr_{B} V (X), C (X) = Tr_{R} V (X) .$
unlabeled · line 301
$d_{tr} (ρ, σ) = \frac{1}{2} ∥ ρ - σ ∥_{1}, d_{⋄} (Φ, Ψ) = \frac{1}{2} ∥ Φ - Ψ ∥_{⋄} .$
unlabeled · line 310
$T_{σ} (X) = Tr (X) σ$

Complete channels and residual response

Displayed formulas in this passage

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