Interchanging limits in the mutual information of two intervals separated by one site

RESUMO

In the paper J. Phys. A: Math. Theor. 53 (2020), 345303, the limiting mutual information between two intervals separated by one lattice site in a free-fermion chain was evaluated. The calculation replaces the entropy function by a regularized version depending on $\varepsilon>0$, takes the block lengths to infinity, and only then lets $\varepsilon$ tend to zero. We prove that the opposite order gives the same constant $2\log2-1$. Moreover, we show that the same limit is obtained whenever both block lengths tend to infinity and $\varepsilon$ tends to zero simultaneously, with no restriction on their relative rates. In fact, the error caused by the regularization is bounded by $(1+\varepsilon)\log(1+\varepsilon)-\varepsilon\log\varepsilon$, uniformly in both block lengths.