Braiding and entanglement in spin networks: a combinatorial approach to topological phases

Details Braiding and entanglement in spin networks: a combinatorial approach to topological phases Braiding and entanglement in spin networks: a combinatorial approach to topological phases

2008-06-24

arxiv.org/abs/0806.3883v1

Int.J.Quant.Inf. 7, Issue 1 supp. (January 2009) 195-203

Physics

Mathematical Physics

Contributed to Quantum 2008: IV workshop ad memoriam of Carlo Novero 19-23 May 2008 - Turin, Italy

Scientific paper

The spin network quantum simulator relies on the su(2) representation ring (or its q-deformed counterpart at q= root of unity) and its basic features naturally include (multipartite) entanglement and braiding. In particular, q-deformed spin network automata are able to perform efficiently approximate calculations of topological invarians of knots and 3-manifolds. The same algebraic background is shared by 2D lattice models supporting topological phases of matter that have recently gained much interest in condensed matter physics. These developments are motivated by the possibility to store quantum information fault-tolerantly in a physical system supporting fractional statistics since a part of the associated Hilbert space is insensitive to local perturbations. Most of currently addressed approaches are framed within a 'double' quantum Chern-Simons field theory, whose quantum amplitudes represent evolution histories of local lattice degrees of freedom. We propose here a novel combinatorial approach based on `state sum' models of the Turaev-Viro type associated with SU(2)_q-colored triangulations of the ambient 3-manifolds. We argue that boundary 2D lattice models (as well as observables in the form of colored graphs satisfying braiding relations) could be consistently addressed. This is supported by the proof that the Hamiltonian of the Levin-Wen condensed string net model in a surface Sigma coincides with the corresponding Turaev-Viro amplitude on Sigma x [0,1] presented in the last section.

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