Manuscript Hawk04-A@1 · Hawk04-A.tex · section · 526092ba

Complete systems and operational definitions

Complete systems and operational definitions · lines 226–272

Hawk04-A@1#eq:channels, Hawk04-A@1#eq:recovery, Hawk04-A@1#eq:leakage

Let $C ≃ C^{d}$ , with $d \geq 2$ , be the logical code. Output Hilbert spaces are separable. States are normal density operators, and channels act on trace-class operators. CPTP means completely positive and trace preserving. Finite-dimensional channel facts and conventions are standard Watrous.

We use

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

The diamond distance retains arbitrary external references. For channels with input $C$ , a reference of dimension $d$ suffices.

For an isometry

V : C ⟶ R \otimes B,

define

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

Here $B$ is the complete inaccessible output. It includes uncollected modes, retained source degrees of freedom, inaccessible apparatus outputs, and preparation purifiers.

For a residual state $σ$ , set

T_{σ} (X) = Tr (X) σ .

Define

ε_{rec} (N) δ_{full} (C) = D CPTP in f d_{⋄} (DN, id_{C}), = σ in f d_{⋄} (C, T_{σ}) .

Displayed formulas in this passage

Hawk04-A@1#eq:channels · line 249
$N (X) = Tr_{B} (V X V^{†}), C (X) = Tr_{R} (V X V^{†}) .$
Hawk04-A@1#eq:recovery · line 264
$ε_{rec} (N) δ_{full} (C) = D CPTP in f d_{⋄} (DN, id_{C}), = σ in f d_{⋄} (C, T_{σ}) .$
Hawk04-A@1#eq:leakage · line 264
$ε_{rec} (N) δ_{full} (C) = D CPTP in f d_{⋄} (DN, id_{C}), = σ in f d_{⋄} (C, T_{σ}) .$
unlabeled · line 236
$d_{tr} (ρ, σ) = \frac{1}{2} ∥ ρ - σ ∥_{1}, d_{⋄} (Φ, Ψ) = \frac{1}{2} ∥ Φ - Ψ ∥_{⋄} .$
unlabeled · line 245
$V : C ⟶ R \otimes B,$
unlabeled · line 260
$T_{σ} (X) = Tr (X) σ .$

Complete systems and operational definitions

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