Differentiate between local and global unitaries


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)?

Mahathi Vempati

Posted 2019-03-01T07:03:38.450

Reputation: 1 878



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.


Posted 2019-03-01T07:03:38.450

Reputation: 35 722