Citation: Zou, Liujun, and Jeongwan Haah. "Spurious long-range entanglement and replica correlation length." Physical Review B 94.7 (2016): 075151.
Web: https://arxiv.org/abs/1604.06101
Tags: Information-theory, Tensor-networks, SPT/SETs, No-go
In this paper, the authors detail a theory of when exactly topological entanglement entropy can go WRONG.
A first major contribution of this paper is to give a fantastic example. This example comes from cluster states on the triangular lattice. They observe that when the cluster state is put on an infinite cylinder, the entanglement between the two halves of the cylinder is (log 2) * L - (log 2) where L is circumference of the cylinder (i.e. number of sites directly above/below the cut). This (log 2) is not just some accident of a bad definition of circumference - it is a real phenomenon which would be called topological entanglement entropy by many numerical algorithms, despite the fact that the product state is topologically trivial!
We now sketch the reason why this (log 2) exists. When considering the entanglement between the two halves of the cylinder, all of the sites away from the cut don't matter - they are only entangled to the neighboring sites and hence can be disentangled by unitaries which are restricted to one side of the cut. This leaves us which a zig-zag chain cluster state around the cut. This zig-zag chain has exactly one stabilizer which is completely localized on one side of the cut - the product of sigma_X operators. This is because the stabilizers in the Hamiltonian for a cluster state are sigma_Z^{j-1} \otimes \sigma_X^{j}\otimes \sigma_Z^{j+1} so when the stabilizers are applied to all of the sites on one side of the cut the \sigma_Z all cancel. These localized stabilizers decrease the entanglement by a factor of (log 2) by general considerations of stabilizer codes.
Of course, this cylinder extrapolation method is weaker than the Kitaev-Preskill prescription. It was first introduced in
> Jiang, Hong-Chen, Zhenghan Wang, and Leon Balents. "Identifying topological order by entanglement entropy." Nature Physics 8.12 (2012): 902-905.
as a DMRG-friendly approach to identifying topological order, and Zhenghan argued that it should be a topological invariant. Clearly, this shows a counterexample.
The counterexample can even be used to trick the original Kitaev-Preskill prescription for TEE, and not just the naive put-your-system-on-a-cylinder approach. This works by choosing your cut, and then putting a zig-zag cluster state on it with half the sites inside and half the sites outside. Just like before, the entanglement is (log 2)*L - (log 2). This correction term is robust, and is measured as TEE by the Kitaev-Preskill method. This counterexample is due to Bravyi, and it motived the present paper. Bravyi's counterexample is very interesting.
The next contribution of this paper is to discuss a theory of why this spurious contribution to TEE appears. They attribute the spurious entanglement in the above example to the fact that the cluster state can be thought of as a Z2\times Z2 SPT with one copy of Z2 acting on each side of the lattice. Moreover, the cohomology class implicit in Wen's classification of SPTs acts nontrivially across the two copies of Z2. Whenever a chain is in the situation that it is an SPT by a product of groups acting on each side of the cut and the cohomology class acts nontrivially across the cut, there will be spurious long-range entanglement. This is a theorem that they prove, giving a large class of examples just like Bravyi's. Of course SPT phases are topologically trivial when the symmetry is ignored and hence should not be given any TEE.
In the case that the chain is not an SPT by a product of groups, it is still possible that there is spurious long-range entanglement. This is attributed to the fact that the so-called "replica correlation length" can be infinite, as discussed in the paper.
The examples discussed in this paper are not the full story. There are other examples of spurious entanglement which violate our beliefs in more fundamental ways:
> Williamson, Dominic J., Arpit Dua, and Meng Cheng. "Spurious topological entanglement entropy from subsystem symmetries." Physical review letters 122.14 (2019): 140506.