  We prove an almost optimal hypercontractive inequality for quantum erasure
channels, generalizing the hypercontractivity for classical binary erasure
channels [NW16]. To our knowledge, this is the first hypercontractivity bound
for non-unital quantum channels. The traditional inductive arguments for
classical hypercontractivity cannot be generalized to the quantum setting due
to the nature of non-commutativity of matrices. To overcome the difficulty, we
establish a multipartite quantum log-Sobolev inequality, which includes the
classical log-Sobolev inequality [DSC96] and the quantum log-Sobolev inequality
[KT13] as one-partite cases. We establish a connection between our multipartite
quantum log-Sobolev inequality and the hypercontractivity bound for quantum
erasure channels via a refined quantum Gross' lemma [Gro75a], extending the
analogous connection [Kin14] between the quantum log-Sobolev inequality and the
hypercontractivity for qubit unital channels. As an application, we prove an
almost tight bound (up to a constant factor) on the classical communication
complexity of two-party common randomness generation assisted with erased-noisy
EPR states, generalizing the tight bound on the same task assisted with
erased-noisy random strings due to Guruswami and Radhakrishnan [GR16].
