Entanglement and area law with a fractal boundary in a topologically ordered phase

Journal Article · Fri Jan 15 04:00:00 EST 2010 · Physical Review. A

DOI:https://doi.org/10.1103/PHYSREVA.81.010102· OSTI ID:21388656

Hamma, Alioscia ^[1]; Lidar, Daniel A ^[2]; Severini, Simone ^[3]

---

1. Perimeter Institute for Theoretical Physics, 31 Caroline St. N, N2L 2Y5, Waterloo Ontario (Canada)
2. Departments of Chemistry, Electrical Engineering, and Physics, and Center for Quantum Information Science and Technology, University of Southern California, Los Angeles, California 90089 (United States)
3. Institute for Quantum Computing and Department of Combinatorics and Optimization University of Waterloo, 200 University Ave. W, N2L 3G1, Waterloo Ontario (Canada)

Quantum systems with short-range interactions are known to respect an area law for the entanglement entropy: The von Neumann entropy S associated to a bipartition scales with the boundary p between the two parts. Here we study the case in which the boundary is a fractal. We consider the topologically ordered phase of the toric code with a magnetic field. When the field vanishes it is possible to analytically compute the entanglement entropy for both regular and fractal bipartitions (A,B) of the system and this yields an upper bound for the entire topological phase. When the A-B boundary is regular we have S/p=1 for large p. When the boundary is a fractal of the Hausdorff dimension D, we show that the entanglement between the two parts scales as S/p=gamma<=1/D, and gamma depends on the fractal considered.

• OSTI ID: 21388656
• Journal Information: Physical Review. A, Journal Name: Physical Review. A Journal Issue: 1 Vol. 81; ISSN 1050-2947; ISSN PLRAAN
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
DISTANCE
ENTROPY
FRACTALS
INTERACTION RANGE
MAGNETIC FIELDS
MATHEMATICS
PHYSICAL PROPERTIES
QUANTUM ENTANGLEMENT
THERMODYNAMIC PROPERTIES
TOPOLOGY