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 |psirangle 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 |psi1,...,nrangle comprised of n qudits is said to be an MPS of bond dimension r if the reduced density matrix psi1,...,k has rank r for each k ∈ 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 |psirangle, and the goal is to determine whether |psirangle is an MPS of bond dimension r or whether |psirangle 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 mathsfQMA(2)=mathsfQMA(k) for k ≥ 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≥ 2, we give an efficient algorithm for testing whether |psirangle is an MPS of bond dimension r using m = O(n r2) copies, independent of the dimensions of the qudits, and we show that Omega(n1/2) 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.