Nearly tight bounds for testing tree tensor network states

TL;DR

This paper establishes nearly tight bounds on the number of copies needed to test whether a quantum state is a tree tensor network state with bounded bond dimension, advancing understanding of quantum state property testing.

Contribution

It provides the first tight bounds for testing TTNS, including both upper and lower bounds, and analyzes measurement restrictions for such testing.

Findings

O(nr^2) copies suffice for testing TTNS with one-sided error.

Omega(n r^2 / log n) copies are necessary when r ≥ 2 + log n.

Theta(√n) copies are needed and sufficient for testing low Schmidt-rank bipartite states.

Abstract

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. We then prove that $\Omega(n r^2/\log n)$ copies are necessary for any test with one-sided error whenever $r\geq 2 + \log n$. In particular, this closes a roughly quadratic gap in the previous bounds for testing matrix product states in this setting. On the other hand, when $r=2$ we show that $\Theta(\sqrt{n})$ copies are both necessary and sufficient for the related task of testing whether a state is a product of $n$ bipartite states having Schmidt-rank at most $r$, for some choice of the qudit dimensions. We also study the performance of tests using measurements performed on a small number of copies at a time. Here, we obtain new bounds for testing rank, Schmidt-rank, and TTNS when the tester is restricted to making measurements on $r+1$ copies of the state.