  We generalize the concepts of weak quantum logarithmic Sobolev inequality
(LSI) and weak hypercontractivity (HC), introduced in the quantum setting by
Olkiewicz and Zegarlinski, to the case of non-primitive quantum Markov
semigroups (QMS). The originality of this work resides in that this new notion
of hypercontractivity is given in terms of the so-called amalgamated
$\mathbb{L}_p$ norms introduced recently by Junge and Parcet in the context of
operator spaces theory. We make three main contributions. The first one is a
version of Gross' integration lemma: we prove that (weak) HC implies (weak)
LSI. Surprisingly, the converse implication differs from the primitive case as
we show that LSI implies HC but with a weak constant equal to the cardinal of
the center of the decoherence-free algebra. Building on the first implication,
our second contribution is the fact that strong LSI and therefore strong HC do
not hold for non-trivially primitive QMS. This implies that the amalgamated
$\mathbb{L}_p$ norms are not uniformly convex for $1\leq p \leq 2$. As a third
contribution, we derive universal bounds on the (weak) logarithmic Sobolev
constants for a QMS on a finite dimensional Hilbert space, using a similar
method as Diaconis and Saloff-Coste in the case of classical primitive Markov
chains, and Temme, Pastawski and Kastoryano in the case of primitive QMS. This
leads to new bounds on the decoherence rates of decohering QMS. Additionally,
we apply our results to the study of the tensorization of HC in non-commutative
spaces in terms of the completely bounded norms (CB norms) recently introduced
by Beigi and King for unital and trace preserving QMS. We generalize their
results to the case of a general primitive QMS and provide estimates on the
(weak) constants.
