Manuscript En06-B@1 · En06-B.tex · section · ee886ae8

Source gate and local projectors

A nontrivial finite quantum instrument › Source gate and local projectors · lines 1296–1326

En06-B@1#eq:projectors, En06-B@1#eq:settingtable

Let

U_{B} = CNOT_{L \to R} (H_{L} \otimes I_{R}), U_{B} ∣ 00 ⟩ = Φ^{+} ⟩ = \frac{∣ 00 ⟩ + ∣ 11 ⟩}{2} .

Write $X, Z$ for Pauli matrices and define

A_{θ} = cos θ Z + sin θ X, P_{a, θ} = \frac{I + a A _{θ}}{2}, a \in {+ 1, - 1} .

Since $A_{θ}^{2} = I$ , the $P_{a, θ}$ are orthogonal projectors with sum $I$ .

Choose local setting tables

θ_{0} = 0, θ_{1} = \frac{π}{2}, φ_{0} = \frac{π}{4}, φ_{1} = - \frac{π}{4} .

Displayed formulas in this passage

unlabeled · line 1299
$U_{B} = CNOT_{L \to R} (H_{L} \otimes I_{R}), U_{B} ∣ 00 ⟩ = Φ^{+} ⟩ = \frac{∣ 00 ⟩ + ∣ 11 ⟩}{2} .$
En06-B@1#eq:projectors · line 1308
$A_{θ} = cos θ Z + sin θ X, P_{a, θ} = \frac{I + a A _{θ}}{2}, a \in {+ 1, - 1} .$
En06-B@1#eq:settingtable · line 1319
$θ_{0} = 0, θ_{1} = \frac{π}{2}, φ_{0} = \frac{π}{4}, φ_{1} = - \frac{π}{4} .$