= Diagonal two-qubit phase entanglement criterion
{title2=$\delta=\phi_{00}-\phi_{01}-\phi_{10}+\phi_{11}$}
A diagonal <two-qubit gate> with entries $e^{i\phi_{jk}}$ is a product of local diagonal gates, up to <global phase>, exactly when $\delta=0$ modulo $2\pi$. This follows by factoring the two-by-two array of phases; its determinant vanishes precisely in that case. If the determinant is nonzero, applying the gate to $|+\rangle\otimes|+\rangle$ produces an entangled state by the <two-qubit product-state determinant criterion>. In particular $\operatorname{diag}(-1,1,1,-1)=-Z\otimes Z$ is local, whereas $\operatorname{diag}(1,1,1,-1)$ is the entangling <Controlled-Z gate>.
Back to article page
