## Differentiate between local and global unitaries

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Just like we have the PPT, NPT criteria for checking if states can be written in tensor form or not, is there any criteria, given the matrix of a unitary acting on 2 qubits, to check if it is local or global (can be factored or not)?

This is actually a much easier problem. In the case of states, you're trying to use the PPT criterion, or others, to distinguish if $$\rho$$ can be written in the form $$\rho=\sum_ip_i\sigma^A_i\otimes\sigma^B_i,$$ where $$\sum_ip_i=1$$ and the $$\sigma^A_i$$ and $$\sigma^B_i$$ are valid states on single sites. The difficulty actually comes from the freedom that the summation over $$i$$ provides.
In the case of unitaries (or more general operations), you're only trying to ascertain if $$U$$ can be written in the form $$U=U^A\otimes U^B$$ or not. This is something that you can do very mechanically. For example, if we make matrices $$\sum_{k,l}U_{ik,jl}|k\rangle\langle l|,$$ then each of these ought to be of the form $$U^A_{ij}U^B$$, in other words, the same up to a constant. If they're not, it's not of tensor product form.