Manuscript Oai211-A@1 · Oai211-A.tex · section · 3f3e4f63

Finite-dimensional channel marks

Finite-dimensional channel marks · lines 602–639

Oai211-A@1#eq:spinechannel, Oai211-A@1#eq:pathchannel

Fix a logical space $C ≅ C^{d}$ , $d \geq 2$ . All channel marks in this section have input and output $C$ . The choice of this finite space is additional operational data, not a consequence of the tree.

For states and channels, use

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

The diamond norm includes arbitrary external references. For any Hermiticity-preserving linear map $Δ$ , write

∥Δ ∥_{⋄, 1/2} = \frac{1}{2} ∥Δ ∥_{⋄} .

This notation also applies to sums that are not channels.

Let $Λ (k, i, A)$ be a fixed measurable CPTP channel-valued function. Define its spine average

Λ = k \geq 1 \sum p_{k} i = 1 \sum^{k} \int Λ (k, i, A) L_{k} (d A) .

The weights sum to one. Finite-dimensional integration and convexity make $Λ$ a CPTP channel.

For $H (v) > 0$ , define

Λ_{v}^{av} = \frac{1}{H ( v )} path (v) \sum Λ (k, i, A) .

Displayed formulas in this passage

Oai211-A@1#eq:spinechannel · line 623
$Λ = k \geq 1 \sum p_{k} i = 1 \sum^{k} \int Λ (k, i, A) L_{k} (d A) .$
Oai211-A@1#eq:pathchannel · line 633
$Λ_{v}^{av} = \frac{1}{H ( v )} path (v) \sum Λ (k, i, A) .$
unlabeled · line 610
$d_{tr} (ρ, σ) = \frac{1}{2} ∥ ρ - σ ∥_{1}, d_{⋄} (Φ, Ψ) = \frac{1}{2} ∥ Φ - Ψ ∥_{⋄} .$
unlabeled · line 616
$∥Δ ∥_{⋄, 1/2} = \frac{1}{2} ∥Δ ∥_{⋄} .$

Finite-dimensional channel marks

Displayed formulas in this passage

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