Nearly tight bounds for testing tree tensor network states

Understand

Tree tensor network states (TTNS) generalize the notion of having low Schmidt-rank to multipartite quantum states, through a parameter known as the bond dimension.

• This leads to succinct representations of quantum many-body systems with a tree-like entanglement structure.
• In this work, we study the task of testing whether an unknown pure state is a TTNS on $n$ qudits with bond dimension at most $r$, or is far in trace distance from any such state.
• We first establish that, independent of the dimension of the state, $O(nr^2)$ copies suffice to accomplish this task with one-sided error.