Uniform bound of the entanglement for the ground state of the quantum Ising model with large transverse magnetic field

We consider the ground state of the quantum Ising model with transverse field $h$ in one dimension in a finite volume $$\mathrm{\Lambda }{}_m:=\{-m,-m+1,\dots ,m+L\}\mathrm{.}$$ For $h$ sufficiently large we prove a bound for the entanglement of the interval $\Lambda_{0}:=\left{ 0,…,L\right} $relativetoitscomplement$\Lambda_{m}\backslash\Lambda_{0}$whichisuniformin$m$and$L$. The bound is established by means of a suitable cluster expansion.