Manuscript Oai290-A@1 · Oai290-A.tex · section · 7c6e104a

A named observable is a different task

A named observable is a different task · lines 512–531

Oai290-A@1#eq:measurementchannel, Oai290-A@1#eq:effecterror

The adjoint $Φ^{*}$ of a channel is defined by $Tr (Φ (X) Y) = Tr (X Φ^{*} (Y))$ . It is unital and completely positive.

Let $F \in B (C)$ be an effect, $0 ⪯ F ⪯ I_{C}$ . Its binary measurement channel is

M_{F} (X) = Tr (F X) ∣ 1 ⟩ ⟨ 1 ∣ + Tr ((I_{C} - F) X) ∣ 0 ⟩ ⟨ 0 ∣ .

A boundary effect $G \in B (A)$ , $0 ⪯ G ⪯ I_{A}$ , reconstructs this readout with error

η_{F} (G) = ∥ N^{*} (G) - F ∥_{\infty} .

Displayed formulas in this passage

Oai290-A@1#eq:measurementchannel · line 520
$M_{F} (X) = Tr (F X) ∣ 1 ⟩ ⟨ 1 ∣ + Tr ((I_{C} - F) X) ∣ 0 ⟩ ⟨ 0 ∣ .$
Oai290-A@1#eq:effecterror · line 527
$η_{F} (G) = ∥ N^{*} (G) - F ∥_{\infty} .$