  We study boundary renormalization group flows between boundary conformal
field theories in $1+1$ dimensions using methods of quantum information theory.
We define an entropic $g$-function for theories with impurities in terms of the
relative entanglement entropy, and we prove that this $g$-function decreases
along boundary renormalization group flows. This entropic $g$-theorem is valid
at zero temperature, and is independent from the $g$-theorem based on the
thermal partition function. We also discuss the mutual information in boundary
RG flows, and how it encodes the correlations between the impurity and bulk
degrees of freedom. Our results provide a quantum-information understanding of
(boundary) RG flow as increase of distinguishability between the UV fixed point
and the theory along the RG flow.
