How to check if a two-qubit gate is entangling?

Question

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...

Answer 1

The metric you're looking for is known as entangling power.

Here are some references:

• Entangling power of two-qubit gates on mixed states (Guan et al., 2014)

• Entangling power of two-qubit unitary operations (Yi Shen & Lin Chen, 2018)

I will expand on this later.