  We consider the problem of entanglement-assisted one-shot classical
communication. In the zero-error regime, entanglement can increase the one-shot
zero-error capacity of a family of classical channels following the strategy of
Cubitt et al., Phys. Rev. Lett. 104, 230503 (2010). This strategy uses the
Kochen-Specker theorem which is applicable only to projective measurements. As
such, in the regime of noisy states and/or measurements, this strategy cannot
increase the capacity. To accommodate generically noisy situations, we examine
the one-shot success probability of sending a fixed number of classical
messages. We show that preparation contextuality powers the quantum advantage
in this task, increasing the one-shot success probability beyond its classical
maximum. Our treatment extends beyond Cubitt et al. and includes, for example,
the experimentally implemented protocol of Prevedel et al., Phys. Rev. Lett.
106, 110505 (2011). We then show a mapping between this communication task and
a corresponding nonlocal game. This mapping generalizes the connection with
pseudotelepathy games previously noted in the zero-error case. Finally, after
motivating a constraint we term context-independent guessing, we show that
contextuality witnessed by noise-robust noncontextuality inequalities obtained
in R. Kunjwal, Quantum 4, 219 (2020), is sufficient for enhancing the one-shot
success probability. This provides an operational meaning to these inequalities
and the associated hypergraph invariant, the weighted max-predictability,
introduced in R. Kunjwal, Quantum 3, 184 (2019). Our results show that the task
of entanglement-assisted one-shot classical communication provides a fertile
ground to study the interplay of the Kochen-Specker theorem, Spekkens
contextuality, and Bell nonlocality.
