Manuscript II-B@1 · II-B-v1.tex · section · bfd2eb3c

Complete binary instruments and centered response

An exact finite response-table instrument › Complete binary instruments and centered response · lines 1726–1752

II-B@1#eq:binary, II-B@1#eq:centered

Destructively read $Q$ in the $Z$ basis and retain the binary value $y \in {+ 1, - 1}$ in a classical register $Y$ . The complete resulting instrument is

B_{f} (ρ) = k, y \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes ∣ y ⟩ ⟨ y ∣_{Y} \otimes \frac{1 + y f ( δ ( k )) / B _{*}}{2} σ_{k} .

Define its centered response map

Δ_{f} = B_{f} - B_{0} .

Thus

Δ_{f} (ρ) = k, y \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes ∣ y ⟩ ⟨ y ∣_{Y} \otimes \frac{y f ( δ ( k ))}{2 B _{*}} σ_{k} .

A centered response map is a linear difference of channels, not generally a completely positive map.

Displayed formulas in this passage

II-B@1#eq:binary · line 1731
$B_{f} (ρ) = k, y \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes ∣ y ⟩ ⟨ y ∣_{Y} \otimes \frac{1 + y f ( δ ( k )) / B _{*}}{2} σ_{k} .$
II-B@1#eq:centered · line 1743
$Δ_{f} (ρ) = k, y \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes ∣ y ⟩ ⟨ y ∣_{Y} \otimes \frac{y f ( δ ( k ))}{2 B _{*}} σ_{k} .$
unlabeled · line 1739
$Δ_{f} = B_{f} - B_{0} .$

Complete binary instruments and centered response

Displayed formulas in this passage

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