Perturbative analysis of topological entanglement entropy from conditional independence

We use the structure of conditionally independent states to analyze the stability of topological entanglement entropy. For the ground state of the quantum double or Levin-Wen model, we obtain a bound on the first-order perturbation of topological entanglement entropy in terms of its energy gap and subsystem size. The bound decreases superpolynomially with the size of the subsystem, provided the energy gap is nonzero. We also study the finite-temperature stability of stabilizer models, for which we prove a stronger statement than the strong subadditivity of entropy. Using this statement and assuming (i) finite correlation length and (ii) small conditional mutual information of certain configurations, first-order perturbation effect for arbitrary local perturbation can be bounded. We discuss the technical obstacles in generalizing these results.

Figure 1

The shaded region represents an effect of the perturbation that is smeared out in space. We shall approximate this effect by a strictly local operator with a finite radius $R$. The correction decreases superpolynomially with $R$.

Figure 2

Applying the isolation move, the conditional entanglement spectrum is deformed in such a way that (i) for the new conditional entanglement spectrum, $O$ is sufficiently far away from the reference party, and (ii) the difference is a conditional entanglement spectrum with small conditional mutual information.

Figure 3

Applying the separation move, the conditional entanglement spectrum is deformed in such a way that (i) for the new conditional entanglement spectrum, $O$ is sufficiently far away from both the reference and target parties, and ( ii) the difference is a conditional entanglement spectrum with small conditional mutual information.

Figure 4

Applying the absorption move, the conditional entanglement spectrum is expressed in terms of a linear combination of conditional entanglement spectrum ${\widehat{H}}_{A_i:C_i|B_i}$ such that (i) the support of $O$ is contained in either $A_i{B}_i$ or $B_i{C}_i$. (ii) $I(A_i:C_i|B_i)$ is small.

Figure 5

We have numerically computed $I(A:C|B)$ and $|{\text{Tr}}_C{\rho }_{ABC}{\widehat{H}}_{A:C|B}{|}_1$ for ${10}^6$ randomly generated pure states. Largest observed ratio $|{\text{Tr}}_C{\rho }_{ABC}{\widehat{H}}_{A:C|B}{|}_1/I(A:C|B)$ was $24.21924$.