Quantum States far from the Energy Eigenstates of Any Local Hamiltonian
Henry L. Haselgrove, Michael A. Nielsen, Tobias J. Osborne · Physical Review Letters · 2003
What quantum states are possible energy eigenstates of a many-body Hamiltonian? Suppose the Hamiltonian is nontrivial, i.e., not a multiple of the identity, and $L$ local, in the sense of containing interaction terms involving at most $L$ bodies, for some fixed $L$. We construct quantum states $\ensuremath{\psi}$ which are ``far away'' from all the eigenstates $\mathbf{E}$ of any nontrivial $L$-local Hamiltonian, in the sense that $\ensuremath{\Vert}\ensuremath{\psi}\ensuremath{-}\mathbf{E}\ensuremath{\Vert}$ is greater than some constant lower bound, independent of the form of the Hamiltonian.