Quantum Codes Give Counterexamples to the Unique Preimage Conjecture of the N-Representability Problem

• Main Authors: Ocko, Samuel Alan (Contributor), Chen, Xie (Contributor), Zeng, Bei (Author), Yoshida, Beni (Author), Ji, Zhengfeng (Author), Ruskai, Mary Beth (Author), Chung, Isaac L. (Author)
• Other Authors: Massachusetts Institute of Technology. Department of Physics (Contributor), Chuang, Isaac L. (Contributor)
• Format: Article
• Language: English
• Published: American Physical Society, 2011-10-03T13:41:59Z.
• Subjects: Article
• Online Access: Get fulltext

 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.