  A fundamental objective in quantum information science is to determine the
cost in classical resources of simulating a particular quantum system. The
classical simulation cost is quantified by the signaling dimension which
specifies the minimum amount of classical communication needed to perfectly
simulate a channel's input-output correlations when unlimited shared randomness
is held between encoder and decoder. This paper provides a collection of
device-independent tests that place lower and upper bounds on the signaling
dimension of a channel. Among them, a single family of tests is shown to
determine when a noisy classical channel can be simulated using an amount of
communication strictly less than either its input or its output alphabet size.
In addition, a family of eight Bell inequalities is presented that completely
characterize when any four-outcome measurement channel, such as a Bell
measurement, can be simulated using one communication bit and shared
randomness. Finally, we bound the signaling dimension for all partial replacer
channels in $d$ dimensions. The bounds are found to be tight for the special
case of the erasure channel.
