MPS Stability and the Intersection Property

TL;DR

This paper introduces a new local property called stability for matrix product states (MPS) tensors, which guarantees the intersection property of their parent Hamiltonians and bounds ground state degeneracy.

Contribution

The paper defines the stability property for MPS tensors, generalizing injectivity, and shows it ensures the intersection property and bounds degeneracy.

Findings

Stability guarantees the intersection property for MPS parent Hamiltonians.

Stability is satisfied by MPS tensors for W states and domain wall states.

Ground space degeneracy is bounded by the square of the bond dimension.

Abstract

We identify a property of the local tensors of matrix product states (MPS) that guarantees that their parent Hamiltonians satisfy the intersection property. The intersection property ensures that the ground space consists of MPS, with degeneracy bounded by the square of the bond dimension. The new local property, dubbed stability, generalizes (block) injectivity and is satisfied by the MPS tensors that construct the W state, domain wall superposition states, and their generalizations.