  We give a simple argument to obtain $\mathrm{L}^p$-boundedness for heat
semigroups associated to uniformly strongly elliptic systems on $\mathbb{R}^d$
by using Stein interpolation between Gaussian estimates and hypercontractivity.
Our results give $p$ explicitly in terms of ellipticity. It is optimal at the
endpoint $p=\infty$. We also obtain $\mathrm{L}^p$-estimates for the gradient
of the semigroup, where $p>2$ depends on ellipticity but not on dimension.
