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

Key preparation

E1: explicit internal and emission Hamiltonians › Key preparation · lines 402–428

Em03-E@1#eq:keypurification

Prepare the key purification by applying Hadamard gates to $K_{a}, K_{b}$ , followed by

CNOT_{K_{a} \to C_{a}}, CNOT_{K_{b} \to C_{b}} .

The resulting state is

∣ κ ⟩_{K C} = \frac{1}{2} a, b = 0 \sum^{1} ∣ a, b ⟩_{K} ∣ a, b ⟩_{C} .

Tracing $C = C_{a} C_{b}$ gives $I_{K} /4$ .

These are actual finite controls. For example, Hadamard is generated exactly by a pulse of area $π$ multiplying $(I - H) /2$ , and a controlled-NOT is generated by a pulse of area $π$ multiplying

∣ 1 ⟩ ⟨ 1 ∣_{control} \otimes \frac{I - X}{2} .

These matrices act within zero-energy sectors.

Displayed formulas in this passage

Em03-E@1#eq:keypurification · line 412
$∣ κ ⟩_{K C} = \frac{1}{2} a, b = 0 \sum^{1} ∣ a, b ⟩_{K} ∣ a, b ⟩_{C} .$
unlabeled · line 406
$CNOT_{K_{a} \to C_{a}}, CNOT_{K_{b} \to C_{b}} .$
unlabeled · line 424
$∣ 1 ⟩ ⟨ 1 ∣_{control} \otimes \frac{I - X}{2} .$