4

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)?

4

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)?

6

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.