Manuscript Em03-E@1 · Em03-E-v1.tex · section · c4b07e6a

Dephasing control

E1: explicit internal and emission Hamiltonians › Dephasing control · lines 429–457

Em03-E@1#eq:dephasingisometry, Em03-E@1#eq:Lambdaeta

Before encryption, apply

U_{ϕ} = exp (- i ϕ ∣ 1 ⟩ ⟨ 1 ∣_{Q} \otimes X_{Z_{e}}), 0 \leq ϕ \leq π /2.

Define

η = cos ϕ .

On the initial environment this gives the isometry

W_{η} ∣ x ⟩ = ∣ x ⟩ ∣ e_{x} ⟩, ∣ e_{0} ⟩ = ∣ 0 ⟩, ∣ e_{1} ⟩ = η ∣ 0 ⟩ - i 1 - η^{2} ∣ 1 ⟩ .

The reduced source channel is

Λ_{η} (X) = η X + (1 - η) Δ_{Z} (X), Δ_{Z} (X) = x = 0 \sum^{1} ∣ x ⟩ ⟨ x ∣ X ∣ x ⟩ ⟨ x ∣ .

This is verified by the overlap $⟨ e_{1} ∣ e_{0} ⟩ = η$ .

Displayed formulas in this passage

Em03-E@1#eq:dephasingisometry · line 442
$W_{η} ∣ x ⟩ = ∣ x ⟩ ∣ e_{x} ⟩, ∣ e_{0} ⟩ = ∣ 0 ⟩, ∣ e_{1} ⟩ = η ∣ 0 ⟩ - i 1 - η^{2} ∣ 1 ⟩ .$
Em03-E@1#eq:Lambdaeta · line 449
$Λ_{η} (X) = η X + (1 - η) Δ_{Z} (X), Δ_{Z} (X) = x = 0 \sum^{1} ∣ x ⟩ ⟨ x ∣ X ∣ x ⟩ ⟨ x ∣ .$
unlabeled · line 432
$U_{ϕ} = exp (- i ϕ ∣ 1 ⟩ ⟨ 1 ∣_{Q} \otimes X_{Z_{e}}), 0 \leq ϕ \leq π /2.$
unlabeled · line 438
$η = cos ϕ .$