Let $U$ be a unitary matrix acting on a 3-qubit system. If there is no correlation among any pairs of the three qubits, the unitary operation can be represented as $U = U_1 \otimes U_2 \otimes U_3$, where $U_i$ exclusively acts on the $i$-th qubit. However, this tensor-product structure does not hold true when $U$ is a highly entangling operation that induces entanglement across all three qubits.

My question is related to the existence of a metric or function that quantifies the degree of entanglement induced by $U$ between specific pairs of qubits. For instance, in the aforementioned scenario, how can one measure how much the given unitary $U$ entangles qubits 1 and 2, as well as qubits 2 and 3?

There are several popular metrics that people use in order to estimate how close the two unitaries are to each other. I am curious to know if there's an analogous metric for assessing the "1-locality" of a given unitary $U$. My naive attempt involves defining a norm in the following manner:

\begin{equation} \min_{U1, U2, U3} \Vert U - U_1 \otimes U_2 \otimes U_3 \Vert, \end{equation} which becomes $0$ if and only if $U$ is indeed a tensor product of $1$-local unitaries. However, this is an optimization across all unitary operations. Therefore, I suspect that more effective metrics are commonly used by people. I would be grateful if someone could discuss these metrics and point me to relevant papers on the subject.

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    $\begingroup$ Why start with three qubits? For two qubits it is less ambiguous how to best quantify entanglement; with three, there are multiple bipartitions to consider, etc. I think it's still an interesting question for two qubits and will help. Do you want to average over all qubit states? Maximize or minimize over qubit states? Evaluate this for a particular state? $\endgroup$ Commented Dec 4, 2023 at 18:42
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    $\begingroup$ I am thinking of taking $\int dV E(U V|\psi\rangle)$ for some entanglement measure $E()$ and some fiducial starting state $|\psi\rangle$ then averaging over all starting states $V|\psi\rangle$. Maybe you'd want to average over all separable starting states, like $v_1|\psi_1\rangle\otimes v_2 |\psi_2\rangle$. And it would be very nice if this average (or its max or its min) could be related to the norm you have written $\endgroup$ Commented Dec 4, 2023 at 18:45


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