If Alice and Bob share only classical communication resources such as noisy or perfect classical channels, is shared entanglement always equivalent to shared randomness?
In other words, must there be a quantum channel in order to exploit the power of shared entanglement? If yes, how can one prove this?
As @Rammus has mentioned in the comments one does not need a quantum channel to utilize entanglement as a resource. One can utilize quantum correlations to aptly perform tasks that are impossible to perform with classical correlations (i.e. shared randomness). For example in a nonlocal game, two players share entanglement (but don't need any kind of quantum channel) and they can use their entanglement to satisfy certain constraints that are impossible with only classical corrlations.
Entanglement + a classical channels allows you to build a quantum channel using teleportation. Thus, adding a quantum channel does not give you additional power.
