  We prove the Shannon's inequality on non-collapsing $\mathsf{RCD}(0,N)$
spaces. In the proof, we use the characterization of the
$\mathsf{EVI}_{0,N}$-gradient flow of the relative entropy and the
infinitesimal behavior of the heat kernel. Also we have a cone rigidity result.
As an application, we have the so-called uncertainty principle inequality on
such spaces.
