title: Quantum codes give counterexamples to the unique preimage conjecture of the N-representability problem.
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 preimage, 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.
authors.affiliation: Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA.
authors.forename: Samuel A
authors.initials: SA
authors.lastname: Ocko
authors.forename: Xie
authors.initials: X
authors.lastname: Chen
authors.forename: Bei
authors.initials: B
authors.lastname: Zeng
authors.forename: Beni
authors.initials: B
authors.lastname: Yoshida
authors.forename: Zhengfeng
authors.initials: Z
authors.lastname: Ji
authors.forename: Mary Beth
authors.initials: MB
authors.lastname: Ruskai
authors.forename: Isaac L
authors.initials: IL
authors.lastname: Chuang