  Understanding the classical communication cost of simulating a quantum
channel is a fundamental problem in quantum information theory, which becomes
even more intriguing when considering the role of non-locality in quantum
information processing. This paper investigates the bidirectional classical
communication cost of simulating a bipartite quantum channel assisted by
non-signalling correlations. Such non-signalling correlations are permitted not
only across spatial dimension between the two parties but also along the
temporal dimension of the channel simulation protocol. By introducing
non-signalling superchannels, we derive semidefinite programming (SDP)
formulations for the one-shot exact bidirectional classical communication cost
via non-signalling bipartite superchannels. We further introduce a channel's
bipartite conditional min-entropy as an efficiently computable lower bound on
the asymptotic cost of bidirectional classical communication. Our results in
both one-shot and asymptotic settings provide lower bounds on the
entanglement-assisted simulation cost in scenarios where entanglement is
available to the two parties and can be utilized across the timeline of the
protocol. Numerical experiments demonstrate the effectiveness of our bounds in
estimating communication costs for various quantum channels, showing that our
bounds can be tight in different scenarios. Our results elucidate the role of
non-locality in quantum communication and pave the way for exploring quantum
reverse Shannon theory in bipartite scenarios.
