  We study conformal field theories with boundaries, and their boundary
renormalization group (RG) flows, using methods from quantum information
theory. Positivity of the relative entropy, together with unitarity and Lorentz
invariance, give rise to bounds that characterize the irreversibility of such
flows. This generalizes the recently proved entropic $g$-theorem to higher
dimensions. In $2+1$ dimensions with a boundary, we prove the entropic
$b$-theorem -- the decrease of the two-dimensional Weyl anomaly under boundary
RG flows. In higher dimensions, the bound implies that the leading area
coefficient of the entanglement entropy induced by the defect decreases along
the flow. Our proof unifies these properties, and provides an
information-theoretic interpretation in terms of the distinguishability between
the short distance and long distance states. Finally, we establish a sum rule
for the change in the area term in theories with boundaries, which could have
implications for models with localized gravity.
