  We prove that every GNS-symmetric quantum Markov semigroup on a finite
dimensional matrix algebra satisfies a modified log-Sobolev inequality. In the
discrete time setting, we prove that every finite dimensional GNS-symmetric
quantum channel satisfies a strong data processing inequality with respect to
its decoherence free part. Moreover, we establish the first general approximate
tensorization property of relative entropy. This extends the famous strong
subadditivity of the quantum entropy (SSA) of two subsystems to the general
setting of two subalgebras. All the three results are independent of the size
of the environment and hence satisfy the tensorization property. They are
obtained via a common, conceptually simple method for proving entropic
inequalities via spectral or $L_2$-estimates. As applications, we combine our
results on the modified log-Sobolev inequality and approximate tensorization to
derive bounds for examples of both theoretical and practical relevance,
including representation of sub-Laplacians on $\operatorname{SU}(2)$ and
various classes of local quantum Markov semigroups such as quantum Kac
generators and continuous time approximate unitary designs. For the latter, our
bounds imply the existence of local continuous time Markovian evolutions on
$nk$ qudits forming $\epsilon$-approximate $k$-designs in relative entropy for
times scaling as $\widetilde{\mathcal{O}}(n^2 \operatorname{poly}(k))$.
