Manuscript En06-B@1 · En06-B.tex · section · ee886ae8

Complete instruments

Geometry, instruments, and admitted networks › Complete instruments · lines 214–242

En06-B@1#eq:completeinstrument

For finite-dimensional Hilbert spaces, an instrument is a finite family of completely positive maps

{Φ_{a} : B (H_{in}) ⟶ B (H_{out})}_{a \in A}

such that $\sum_{a} Φ_{a}$ is trace preserving. Its complete record-retaining channel is

Φ (X) = a \in A \sum Φ_{a} (X) \otimes ∣ a ⟩ ⟨ a ∣ .

Classical registers are represented by their diagonal matrix algebras. When a full-matrix extension is needed, input classical registers are dephased in their declared bases.

We compare complete channels using

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

The diamond norm and finite-dimensional channel representations are standard; see Watrous Watrous. The instrument distinction originates in the operational framework of Davies and Lewis DaviesLewis.

Displayed formulas in this passage

unlabeled · line 218
${Φ_{a} : B (H_{in}) ⟶ B (H_{out})}_{a \in A}$
En06-B@1#eq:completeinstrument · line 224
$Φ (X) = a \in A \sum Φ_{a} (X) \otimes ∣ a ⟩ ⟨ a ∣ .$
unlabeled · line 234
$d_{⋄} (Φ, Ψ) = \frac{1}{2} ∥ Φ - Ψ ∥_{⋄} .$

Complete instruments

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