The Structure of Matrix Product States

For the past twenty years, Matrix Product States (MPS) have been widely used in solid state physics to approximate the ground state of one-dimensional spin chains. In this Letter, exploiting an unnoticed connection with the theory of matrix algebras, we derive two structural properties of MPS, namely: a) there exist local operators which are invisible for all MPS of a given bond dimension; and b) there exist local transformations which, when applied over any MPS of a given bond dimension, decouple (cut) the particles where they act from the spin chain while at the same time join (glue) the two loose ends back again into an MPS. We use the notions of `invisible' and `cut-and-glue' operators to explore the limitations of the MPS approximation. In addition, we show how the high dimensionality of the `invisible space' leads to an exponential decrease in complexity of the SDP hierarchies described in [Phys. Rev. Lett. 115, 020501 (2015)] for the characterization of finite-dimensional quantum correlations. Finally, we generalize some of our results to the ansatz of Projected Entangled Pairs States (PEPS).