Quantum codes give counterexamples to the unique pre-image conjecture of the N-representability problem

          @misc{ pirsa_PIRSA:11020102,
            doi = {10.48660/11020102},
            url = {https://pirsa.org/11020102},
            author = {Ocko, Sam},
            keywords = {Quantum Information},
            language = {en},
            title = {Quantum codes give counterexamples to the unique pre-image conjecture of the N-representability problem},
            publisher = {Perimeter Institute},
            year = {2011},
            month = {feb},
            note = {PIRSA:11020102 see, \url{https://pirsa.org}}
          }
          

Abstract

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.