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Quantum codes give counterexamples to the unique preimage conjecture of the N-representability problem
Samuel A Ocko1, Xie Chen, Bei Zeng
1Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA.
Physical Review Letters
|April 8, 2011
Summary
Researchers found counterexamples to a 1972 conjecture about N-representable two-particle density matrices. This challenges the uniqueness of N-particle preimages for ground states of 2-body Hamiltonians.
Area of Science:
- Quantum Many-Body Physics
- Quantum Information Theory
Background:
- Ground state energy calculations for many-particle systems often use reduced density matrices.
- A key challenge is determining N-representability of these matrices.
- A 1972 conjecture proposed unique N-particle preimages for extreme N-representable two-particle density matrices.
Purpose of the Study:
- To investigate the validity of the 1972 conjecture regarding N-representability.
- To explore the relationship between N-representability, ground state degeneracy, and quantum error correction.
Main Methods:
- Constructing specific Hamiltonians with 2-body interactions.
- Analyzing degenerate ground states that are indistinguishable by 2-body operators.
- Connecting counterexamples to concepts from quantum error correction codes and topological order.
Main Results:
- Explicit counterexamples to the 1972 conjecture were presented.
- Degenerate ground states were identified that lack unique N-particle preimages.
- The existence of these counterexamples is linked to quantum error correction and topological spin systems.
Conclusions:
- The 1972 conjecture regarding unique N-particle preimages for extreme N-representable two-particle density matrices is not universally true.
- The study highlights a deeper connection between the properties of reduced density matrices and fundamental concepts in quantum information and condensed matter physics.
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