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Topological Entanglement

Steven H. Simon

Published Sep 29, 2023• 0 citations
Publisher

Abstract

Abstract If we partition a Hilbert space we can define the entanglement entropy (both von Neumann and Renyi) between the two pieces. We consider breaking up a system into two or more distinct spatial regions and ask about the entanglement between these two regions. Studying the toric code ground state as an initial example, we show that there is an entanglement that scales as the length of the cut between the two regions, but there is also a sub-leading term, known as the topological entanglement entropy, which depends only on the topological properties of the system. While the leading term is nonuniversal, the topological term is robust and can be generalized to any other TQFT. We give three different derivations of the topological entanglement entropy.

DOI
10.1093/oso/9780198886723.003.0034

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Abstract If we partition a Hilbert space we can define the entanglement entropy (both von Neumann and Renyi) between the two pieces. We consider breaking up a system into two or more distinct spatial regions and ask about the entanglement between these two regions. Studying the toric code ground state as an initial example, we show that there is an entanglement that scales as the length of the cut between the two regions, but there is also a sub-leading term, known as the topological entanglement entropy, which depends only on the topological properties of the system. While the leading term is nonuniversal, the topological term is robust and can be generalized to any other TQFT. We give three different derivations of the topological entanglement entropy.
The entanglement properties of a class of topological stabilizer states, the so-called topological color codes defined on a two-dimensional lattice or 2-colex, are calculated. The topological entropy is used to measure the entanglement of different bipartitions of the 2-colex. The dependency of the ground-state degeneracy on the genus of the surface shows that the color code can support a topological order, and the contribution of the color in its structure makes it interesting to compare with Kitaev's toric code. While a qubit is maximally entangled with the rest of the system, two qubits are no longer entangled showing that the color code is genuinely multipartite entangled. For a convex region, it is found that entanglement entropy depends only on the degrees of freedom living on the boundary of two subsystems. The boundary scaling of entropy is supplemented with a topological subleading term, which for a color code defined on a compact surface, is twice over the toric code. From the entanglement entropy we construct a set of bipartitions in which the diverging term arising from the boundary term is washed out, and the remaining nonvanishing term will have a topological nature. Besides the color code on the compact surface, we also analyze the entanglement properties of a version of color code with a border, i.e., triangular color code.
We discuss entanglement entropy of gapped ground states in different dimensions, obtained on partitioning space into two regions. For trivial phases without topological order, we argue that the entanglement entropy may be obtained by integrating an ``entropy density'' over the partition boundary that admits a gradient expansion in the curvature of the boundary. This constrains the expansion of entanglement entropy as a function of system size and points to an even-odd dependence on dimensionality. For example, in contrast to the familiar result in two dimensions, a size-independent constant contribution to the entanglement entropy can appear for trivial phases in any odd spatial dimension. We then discuss phases with topological entanglement entropy (TEE) that cannot be obtained by adding local contributions. We find that in three dimensions there is just one type of TEE, as in two dimensions, that depends linearly on the number of connected components of the boundary (the ``zeroth Betti number''). In D>3D>3D>3 dimensions, new types of TEE appear which depend on the higher Betti numbers of the boundary manifold. We construct generalized toric code models that exhibit these TEEs and discuss ways to extract TEE in $D\ensuremath{\ge}3$.
Abstract We compute the topological entanglement entropy for a large set of lattice models in d-dimensions. It is well known that many such quantum systems can be constructed out of lattice gauge models. For dimensionality higher than two, there are generalizations going beyond gauge theories, which are called higher gauge theories and rely on higher-order generalizations of groups. Our main concern is a large class of d-dimensional quantum systems derived from Abelian higher gauge theories. In this paper, we derive a general formula for the bipartition entanglement entropy for this class of models, and from it we extract both the area law and the sub-leading terms, which explicitly depend on the topology of the entangling surface. We show that the entanglement entropy SA in a sub-region A is proportional to log (GSDA~) \left({GSD}_{\tilde{A}}\right) (GSDA~​) GSD A ˜ , where GSDA~ {GSD}_{\tilde{A}} GSDA~​ GSD A ˜ is the ground state degeneracy of a particular restriction of the full model to A. The quantity GSDA~ {GSD}_{\tilde{A}} GSDA~​ GSD A ˜ can be further divided into a contribution that scales with the size of the boundary ∂A and a term which depends on the topology of ∂A. There is also a topological contribution coming from A itself, that may be non-zero when A has a non-trivial homology. We present some examples and discuss how the topology of A affects the topological entropy. Our formalism allows us to do most of the calculation for arbitrary dimension d. The result is in agreement with entanglement calculations for known topological models.
A bstract The topological Rényi and entanglement entropies depend on the bipartition of the manifold and the choice of the ground states. However, these entanglement quantities remain invariant under a coordinate transformation when the bipartition also undergoes the identical transformation. In topological quantum field theories (TQFTs), these coordinate transformations reduce to representations of the mapping class group on the manifold of the Hilbert space. We employ this invariant property of the Rényi and entanglement entropies under coordinate transformations for TQFTs in (2 + 1) dimensions on a torus with various bipartitions. By utilizing the replica trick and the surgery method to compute the topological Rényi and entanglement entropies, the invariant property results in Verlinde-like formulas. Furthermore, for the bipartition with interfaces as two non-intersecting torus knots, an SL(2 , ℤ) transformation can untwist the torus knots, leading to a simple bipartition with an effective ground state. This invariant property allows us to demonstrate that the topological entanglement entropy has a lower bound −2 ln D , where D is the total quantum dimensions of the system.
We elucidate the topological features of the entanglement entropy of a region in two-dimensional quantum systems in a topological phase with a finite correlation length $\ensuremath{\xi}$. First, we suggest that simpler reduced quantities, related to the von Neumann entropy, could be defined to compute the topological entropy. We use our methods to compute the entanglement entropy for the ground-state wave function of a quantum eight-vertex model in its topological phase and show that a finite correlation length adds corrections of the same order as the topological entropy which come from sharp features of the boundary of the region under study. We also calculate the topological entropy for the ground state of the quantum dimer model on a triangular lattice by using a mapping to a loop model. The topological entropy of the state is determined by loop configurations with a nontrivial winding number around the region under study. Finally, we consider extensions of the Kitaev wave function, which incorporate the effects of electric and magnetic charge fluctuations, and use it to investigate the stability of the topological phase by calculating the topological entropy.
A definition for the entanglement entropy in a gauge theory was given recently in arXiv:1501.02593. Working on a spatial lattice, it involves embedding the physical state in an extended Hilbert space obtained by taking the tensor product of the Hilbert space of states on each link of the lattice. This extended Hilbert space admits a tensor product decomposition by definition and allows a density matrix and entanglement entropy for the set of links of interest to be defined. Here, we continue the study of this extended Hilbert space definition with particular emphasis on the case of Non-Abelian gauge theories. We extend the electric centre definition of Casini, Huerta and Rosabal to the Non-Abelian case and find that it differs in an important term. We also find that the entanglement entropy does not agree with the maximum number of Bell pairs that can be extracted by the processes of entanglement distillation or dilution, and give protocols which achieve the maximum bound. Finally, we compute the topological entanglement entropy which follows from the extended Hilbert space definition and show that it correctly reproduces the total quantum dimension in a class of Toric code models based on Non-Abelian discrete groups.