I would like to know if there's an analog for Schmidt rank that can tell me if a two-qubit unitary is entangling?
Suppose I have a parametrized two-qubit unitary $U^{(2)}(\theta)$. I would like to know a test to tell the degree of entanglement this gate produces, ideally as a function of $\theta$. For example, given the following matrix:
$$ A = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & e^{i\theta} & 0 \\ 0 & 0 & 0 & -e^{i\theta} \end{pmatrix} $$
the test should tell me that that $A \propto Z \otimes R_z(\theta)$ is not entangling. But given the matrix: $$ B = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & e^{-i\theta} & 0 \\ 0 & 0 & 0 & e^{i\theta} \end{pmatrix} $$ the test should tell me that $B = CR_z(\theta)$ is entangling (provided $\theta \neq 0$). In the best case, the test would yield some degree of entanglement as a function of $\theta$ though such a metric may depend on the initial state instead of just the gate...
