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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...

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

Here are some references:

I will expand on this later.

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  • $\begingroup$ This is a very interesting metric. I've never heard of this before and will need some time to work through those refs. In the meantime could you provide an overview of whats going on here? $\endgroup$ – forky40 Oct 22 at 22:17

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