A simple mechanism for unstable degeneracies in local Hamiltonians

TL;DR

This paper discusses how local Hamiltonian eigenstates related by commuting local operators cannot both be unique ground states or have protected degeneracies, impacting quantum state stability.

Contribution

It introduces a fundamental no-go result showing limitations on degeneracy protection in local Hamiltonians due to eigenstate relations.

Findings

Eigenstates related by commuting local operators cannot both be unique ground states.

Such eigenstates cannot have degeneracies protected against local perturbations.

The result constrains the design of stable quantum systems.

Abstract

If a local Hamiltonian eigenstate is mapped to another state by local operators commuting with the Hamiltonian terms, the latter is also an eigenstate. This basic observation implies a no-go result for both being a unique ground state and having a degeneracy protected against local perturbations.