Manuscript Oai283-A@1 · Oai283-A.tex · section · 6fb75895

Norm and Choi conventions

Channels, norms, and finite oracle circuits › Norm and Choi conventions · lines 213–246

For finite-dimensional spaces, $B (H)$ denotes the full matrix algebra. All channels are completely positive and trace preserving. For channels $Φ, Ψ$ with the same interfaces, put

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

For input dimension $d$ , define

∣ Ω_{d} ⟩ = \frac{1}{d} j = 0 \sum^{d - 1} ∣ j ⟩ ∣ j ⟩, J_{n} (Φ) = (id_{d} \otimes Φ) (∣ Ω_{d} ⟩ ⟨ Ω_{d} ∣),

and the unnormalized, input-first Choi matrix

J (Φ) = d J_{n} (Φ) = i, j \sum E_{ij} \otimes Φ (E_{ij}) .

Thus

J (Φ) \geq 0, Tr_{out} J (Φ) = I_{in} .

We use the standard characterization of channel diamond distance as the maximum output trace distance over reference-entangled states; a reference of input dimension suffices WatrousBook (Chapter 3). Contractivity under CPTP pre- and post-processing follows immediately from this characterization.

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unlabeled · line 218
$d_{⋄} (Φ, Ψ) = \frac{1}{2} ∥Φ - Ψ ∥_{⋄}, d_{tr} (ρ, σ) = \frac{1}{2} ∥ ρ - σ ∥_{1} .$
unlabeled · line 224
$∣ Ω_{d} ⟩ = \frac{1}{d} j = 0 \sum^{d - 1} ∣ j ⟩ ∣ j ⟩, J_{n} (Φ) = (id_{d} \otimes Φ) (∣ Ω_{d} ⟩ ⟨ Ω_{d} ∣),$
unlabeled · line 231
$J (Φ) = d J_{n} (Φ) = i, j \sum E_{ij} \otimes Φ (E_{ij}) .$
unlabeled · line 236
$J (Φ) \geq 0, Tr_{out} J (Φ) = I_{in} .$

Norm and Choi conventions

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