  We prove that the quantum relative entropy decreases monotonically under the
application of any positive trace-preserving linear map, for underlying
separable Hilbert spaces. This answers in the affirmative a natural question
that has been open for a long time, as monotonicity had previously only been
shown to hold under additional assumptions, such as complete positivity or
Schwarz-positivity of the adjoint map. The first step in our proof is to show
monotonicity of the sandwiched Renyi divergences under positive
trace-preserving maps, extending a proof of the data processing inequality by
Beigi [J. Math. Phys. 54, 122202 (2013)] that is based on complex interpolation
techniques. Our result calls into question several measures of non-Markovianity
that have been proposed, as these would assess all positive trace-preserving
time evolutions as Markovian.
