Manuscript Psy03-A@1 · Psy03-A.tex · section · b30eb6e9

Phase-sensitive classification under dephasing

Exact coherence-sensitive pilots › Phase-sensitive classification under dephasing · lines 1616–1648

Psy03-A@1#eq:dephchannel

On $C^{2}$ , let

X = ∣ 0 ⟩ ⟨ 1 ∣ + ∣ 1 ⟩ ⟨ 0 ∣, Y = - i ∣ 0 ⟩ ⟨ 1 ∣ + i ∣ 1 ⟩ ⟨ 0 ∣, Z = ∣ 0 ⟩ ⟨ 0 ∣ - ∣ 1 ⟩ ⟨ 1 ∣ .

Define

L_{γ} = \frac{γ}{2} D_{Z}, λ = e^{- γ T} .

Then

Λ_{λ} (ρ) = \frac{1 + λ}{2} ρ + \frac{1 - λ}{2} Z ρZ .

For the equal-prior binary task, encode labels by

ρ_{\pm} = ∣ \pm ⟩ ⟨ \pm ∣, ∣ \pm ⟩ = (∣ 0 ⟩ \pm ∣ 1 ⟩) / 2,

and measure with the complete projective instrument

I_{\pm} (ρ) = P_{\pm} ρ P_{\pm}, P_{\pm} = ∣ \pm ⟩ ⟨ \pm ∣ .

Displayed formulas in this passage

unlabeled · line 1619
$X = ∣ 0 ⟩ ⟨ 1 ∣ + ∣ 1 ⟩ ⟨ 0 ∣, Y = - i ∣ 0 ⟩ ⟨ 1 ∣ + i ∣ 1 ⟩ ⟨ 0 ∣, Z = ∣ 0 ⟩ ⟨ 0 ∣ - ∣ 1 ⟩ ⟨ 1 ∣ .$
unlabeled · line 1625
$L_{γ} = \frac{γ}{2} D_{Z}, λ = e^{- γ T} .$
Psy03-A@1#eq:dephchannel · line 1630
$Λ_{λ} (ρ) = \frac{1 + λ}{2} ρ + \frac{1 - λ}{2} Z ρZ .$
unlabeled · line 1639
$ρ_{\pm} = ∣ \pm ⟩ ⟨ \pm ∣, ∣ \pm ⟩ = (∣ 0 ⟩ \pm ∣ 1 ⟩) / 2,$
unlabeled · line 1644
$I_{\pm} (ρ) = P_{\pm} ρ P_{\pm}, P_{\pm} = ∣ \pm ⟩ ⟨ \pm ∣ .$