  It is well known that the ground state energy of many-particle Hamiltonians
involving only 2-body interactions can be obtained using constrained
optimizations over density matrices which arise from reducing an N-particle
state. While determining which 2-particle density matrices are "N-
representable" is a computationally hard problem, all known extreme
N-representable 2-particle reduced density matrices arise from a unique
N-particle pre-image, satisfying a conjecture established in 1972. We present
explicit counterexamples to this conjecture through giving Hamiltonians with
2-body interactions which have degenerate ground states that cannot be
distinguished by any 2-body operator. We relate the existence of such
counterexamples to quantum error correction codes and topologically ordered
spin systems.
