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

Samuel A. Ocko, Xie Chen, Bei Zeng, Beni Yoshida, Zhengfeng Ji, Mary Beth Ruskai, and Isaac L. Chuang
Phys. Rev. Lett. 106, 110501 – Published 14 March 2011

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.

  • Figure
  • Received 31 October 2010

DOI:https://doi.org/10.1103/PhysRevLett.106.110501

© 2011 American Physical Society

Authors & Affiliations

Samuel A. Ocko1, Xie Chen1, Bei Zeng2,3, Beni Yoshida1, Zhengfeng Ji4, Mary Beth Ruskai5, and Isaac L. Chuang1

  • 1Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA
  • 2Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada
  • 3Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada
  • 4Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada
  • 5Department of Electrical and Computer Engineering, Tufts University, Medford, Massachusetts, USA

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Issue

Vol. 106, Iss. 11 — 18 March 2011

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