  We derive the monotonicity of the quantum relative entropy by an elementary
operational argument based on Stein's lemma in quantum hypothesis testing. For
the latter we present an elementary and short proof that requires the law of
large numbers only. Joint convexity of the quantum relative entropy is proven
too, resulting in a self-contained elementary version of Tropp's approach to
Lieb's concavity theorem, according to which the map tr(exp(h+log a)) is
concave in a on positive operators for self-adjoint h.
