Universal Lower Bound on Topological Entanglement Entropy

Entanglement entropies of two-dimensional gapped ground states are expected to satisfy an area law, with a constant correction term known as the topological entanglement entropy (TEE). In many models, the TEE takes a universal value that characterizes the underlying topological phase. However, the TEE is not truly universal: it can differ even for two states related by constant-depth circuits, which are necessarily in the same phase. The difference between the TEE and the value predicted by the anyon theory is often called the “spurious” topological entanglement entropy. We show that this spurious contribution is always non-negative, thus the value predicted by the anyon theory provides a universal lower bound. This observation also leads to a definition of TEE that is invariant under constant-depth quantum circuits.

Figure 1

(a) The partition used to calculate the TEE. (b) Support of the unitary $U$ considered. (c) String operator $V_a$ creates an anyon $a$ in the interior of the annulus and its antiparticle in the exterior.

Figure 2

For any constant-depth circuit $U$, for any subsystem $S$, we can obtain a circuit $U^{\prime }$ of same depth acting trivially on $S$, by removing from $U$ the “light cone” (white gates) of $S$.

Figure 3

(a) We first remove gates from $U$ on a “hole” within $A$; we call the new circuit $U^{\prime }$. (b) We then deform $A$ to $A^{\prime }\subset A$ such that the boundary of the annulus $A^{\prime }BC$ is only partially covered. (c) We then further remove some of the gates in the vicinity of $A^{\prime }$, obtaining $U^{\prime \prime }$.

Figure 4

Using the procedure in Fig. 2, we remove the gates in $U$ that act deep in the interior of the annulus without changing the entropy; starting from $U$, whose support is depicted as the blue region in (a), we obtain a unitary $\overline{U}$, whose support is shown in (b). Then the support of $\overline{U}$ becomes a union of two disks, which is topologically equivalent to (c) after regrouping sites.

Figure 5

The construction of $\mathcal{R}$. The first step is the partial trace ${\mathrm{Tr}}_{\overline{u}}$ over the support of $\overline{U}$, and the second step applies the Petz map ${\mathrm{\Phi }}_{v\to v\overline{u}}^{\sigma }$, where $v=v_1{v}_2$ and $\overline{u}={\overline{u}}_1{\overline{u}}_2$. The blue subsystem is $\overline{u}$, the support of $\overline{U}$.