Bond dimension witnesses and the structure of homogeneous matrix product states

TL;DR

This paper explores the structure of homogeneous matrix product states (hMPS), deriving tools to identify their bond dimension and analyzing limitations and computational improvements in their use for quantum many-body problems.

Contribution

It introduces bond dimension witnesses for hMPS, reveals structural properties via matrix algebra, and extends results to PEPS, enhancing understanding and computational methods for quantum states.

Findings

Existence of local operators that annihilate all hMPS of a given bond dimension

Existence of local operators that decouple and glue particles in hMPS

Systematic derivation of bond dimension witnesses for hMPS

Abstract

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 paper, we study homogeneous MPS (hMPS), or MPS constructed via site-independent tensors and a boundary condition. Exploiting a connection with the theory of matrix algebras, we derive two structural properties shared by all hMPS, namely: a) there exist local operators which annihilate all hMPS of a given bond dimension; and b) there exist local operators which, when applied over any hMPS 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 a hMPS. Armed with these tools, we show how to systematically derive `bond dimension witnesses', or 2-local operators whose expectation value allows us to lower bound the bond dimension of the underlying hMPS. We extend some of these results to the ansatz of Projected Entangled Pairs States (PEPS). As a bonus, we use our insight on the structure of hMPS to: a) derive some theoretical limitations on the use of hMPS and hPEPS for ground state energy computations; b) show how to decrease the complexity and boost the speed of convergence of the semidefinite programming hierarchies described in [Phys. Rev. Lett. 115, 020501 (2015)] for the characterization of finite-dimensional quantum correlations.