Geometric Entanglement in Topologically Ordered States

Abstract

Here we investigate the connection between topological order and the geometric entanglement, as measured by the logarithm of the overlap between a given state and its closest product state of blocks. We do this for a variety of topologically-ordered systems such as the toric code, double semion, color code, and quantum double models. As happens for the entanglement entropy, we find that for sufficiently large block sizes the geometric entanglement is, up to possible sub-leading corrections, the sum of two contributions: a bulk contribution obeying a boundary law times the number of blocks, and a contribution quantifying the underlying pattern of long-range entanglement of the topologically-ordered state. This topological contribution is also present in the case of single-spin blocks in most cases, and constitutes an alternative characterisation of topological order for these quantum states based on a multipartite entanglement measure. In particular, we see that the topological term for the 2D color code is twice as much as the one for the toric code, in accordance with recent renormalization group arguments. Motivated by this, we also derive a general formalism to obtain upper- and lower-bounds to the geometric entanglement of states with a non-Abelian symmetry, and which we use to analyse quantum double models. Furthermore, we also analyse the robustness of the topological contribution using renormalization and perturbation theory arguments, as well as a numerical estimation for small systems. Some of our results rely on the ability to disentangle single sites from the system, which is always possible in our framework. Aditionally, we relate our results to the relative entropy of entanglement in topological systems, and discuss some tensor network numerical approaches that could be used to extract the topological contribution for large systems beyond exactly-solvable models.

20 pages, 12 figures, 1 table, revised version to appear in NJP. This paper supersedes the results in arXiv:1108.1537

References in corpus (17)

• Non-Abelian Anyons and Topological Quantum Computation
• A class of quantum many-body states that can be efficiently simulated
• Topological Quantum Distillation
• Criticality, the area law, and the computational power of PEPS
• Measuring entanglement growth in quench dynamics of bosons in an optical lattice
• Valence Bond Solids for Quantum Computation
• Bipartite entanglement and entropic boundary law in lattice spin systems
• Equivalence of critical scaling laws for many-body entanglement in the Lipkin-Meshkov-Glick model
• Entanglement renormalization and topological order
• Resonating valence bond states in the PEPS formalism
• Universal geometric entanglement close to quantum phase transitions
• Entanglement of multiparty stabilizer, symmetric, and antisymmetric states
• Exact entanglement renormalization for string-net models
• Geometric Entanglement in a One-Dimensional Valence Bond Solid State
• Experimental Quantum Simulation of Entanglement in Many-body Systems
• Entanglement properties of topological color codes
• Dressed, noise- or disorder- resilient optical lattices

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