We investigate whether certain non-classical communication channels can be simulated by a classical channel with a given number of states and a given `amount' of noise. It is proved that any noisy quantum channel can be simulated by a corresponding classical channel with `the same amount' of noise. Classical simulations of general probabilistic channels are also studied.
It is easy to see that the classical channel with $n$ states can be simulated by the quantum channel of level $n$. By a theorem of Weiner and the present author, the converse also holds. The present paper is about variants of this theorem for general probabilistic channels and for noisy quantum channels. We also discuss noiseless classical simulations of noisy channels, and present an open problem tentatively linking classical simulations of quantum channels to the more traditional way of comparing efficiency of classical and quantum communication, involving von Neumann entropy, mutual information and Holevo's inequality.
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