  There are various ways to quantify the communication capabilities of a
quantum channel. In this work we study the communication value (cv) of channel,
which describes the optimal success probability of transmitting a randomly
selected classical message over the channel. The cv also offers a dual
interpretation as the classical communication cost for zero-error channel
simulation using non-signaling resources. We first provide an entropic
characterization of the cv as a generalized conditional min-entropy over the
cone of separable operators. Additionally, the logarithm of a channel's cv is
shown to be equivalent to its max-Holevo information, which can further be
related to channel capacity. We evaluate the cv exactly for all qubit channels
and the Werner-Holevo family of channels. While all classical channels are
multiplicative under tensor product, this is no longer true for quantum
channels in general. We provide a family of qutrit channels for which the cv is
non-multiplicative. On the other hand, we prove that any pair of qubit channels
have multiplicative cv when used in parallel. Even stronger, all
entanglement-breaking channels and the partially depolarizing channel are shown
to have multiplicative cv when used in parallel with any channel. We then turn
to the entanglement-assisted cv and prove that it is equivalent to the
conditional min-entropy of the Choi matrix of the channel. A final component of
this work investigates relaxations of the channel cv to other cones such as the
set of operators having a positive partial transpose (PPT). The PPT cv is
analytically and numerically investigated for well-known channels such as the
Werner-Holevo family and the dephrasure family of channels.
