Testing matrix product states

Devising schemes for testing the amount of entanglement in quantum systems has played a crucial role in quantum computing and information theory. Here, we study the problem of testing whether an unknown state $|ψ\rangle$ is a matrix product state (MPS) in the property testing model. MPS are a class of physically-relevant quantum states which arise in the study of quantum many-body systems. A quantum state $|ψ_{1,...,n}\rangle$ comprised of $n$ qudits is said to be an MPS of bond dimension $r$ if the reduced density matrix $ψ_{1,...,k}$ has rank $r$ for each $k \in \{1,...,n\}$. When $r=1$, this corresponds to the set of product states. For larger values of $r$, this yields a more expressive class of quantum states, which are allowed to possess limited amounts of entanglement. In the property testing model, one is given $m$ identical copies of $|ψ\rangle$, and the goal is to determine whether $|ψ\rangle$ is an MPS of bond dimension $r$ or whether $|ψ\rangle$ is far from all such states. For the case of product states, we study the product test, a simple two-copy test previously analyzed by Harrow and Montanaro (FOCS 2010), and a key ingredient in their proof that $\mathsf{QMA(2)}=\mathsf{QMA}(k)$ for $k \geq 2$. We give a new and simpler analysis of the product test which achieves an optimal bound for a wide range of parameters, answering open problems of Harrow and Montanaro (FOCS 2010) and Montanaro and de Wolf (2016). For the case of $r\geq 2$, we give an efficient algorithm for testing whether $|ψ\rangle$ is an MPS of bond dimension $r$ using $m = O(n r^2)$ copies, independent of the dimensions of the qudits, and we show that $Ω(n^{1/2})$ copies are necessary for this task. This lower bound shows that a dependence on the number of qudits $n$ is necessary, in sharp contrast to the case of product states where a constant number of copies suffices.