Physical properties of the ground and excited states of a $k$-local
Hamiltonian are largely determined by the $k$-particle reduced density matrices
($k$-RDMs), or simply the $k$-matrix for fermionic systems---they are at least
enough for the calculation of the ground state and excited state energies.
Moreover, for a non-degenerate ground state of a $k$-local Hamiltonian, even
the state itself is completely determined by its $k$-RDMs, and therefore
contains no genuine ${>}k$-particle correlations, as they can be inferred from
$k$-particle correlation functions. It is natural to ask whether a similar
result holds for non-degenerate excited states. In fact, for fermionic systems,
it has been conjectured that any non-degenerate excited state of a 2-local
Hamiltonian is simultaneously a unique ground state of another 2-local
Hamiltonian, hence is uniquely determined by its 2-matrix. And a weaker version
of this conjecture states that any non-degenerate excited state of a 2-local
Hamiltonian is uniquely determined by its 2-matrix among all the pure
$n$-particle states. We construct explicit counterexamples to show that both
conjectures are false. It means that correlations in excited states of local
Hamiltonians could be dramatically different from those in ground states. We
further show that any non-degenerate excited state of a $k$-local Hamiltonian
is a unique ground state of another $2k$-local Hamiltonian, hence is uniquely
determined by its $2k$-RDMs (or $2k$-matrix).
