$W$ state is not the unique ground state of any local Hamiltonian

Understand

The characterization of ground states among all quantum states is an important problem in quantum many-body physics.

• For example, the celebrated entanglement area law for gapped Hamiltonians has allowed for efficient simulation of 1d and some 2d quantum systems using matrix product states.
• Among ground states, some types, such as cat states (like the GHZ state) or topologically ordered states, can only appear alongside their degenerate partners, as is understood from the theory of spontaneous symmetry breaking.
• In this work, we introduce a new class of simple states, including the $W$ state, that can only occur as a ground state alongside an exactly degenerate partner, even in gapless or disordered models.