A communication protocol with $l$ possible inputs and $k$ possible outputs can be described by a transition matrix $A=(a_{ij})\in [0,1]^{k\times l}$, where $a_{ij}$ is the conditional probability of output $i$ if the input is $j$. This is a stochastic matrix, i.e., all entries are non-negative and each column sums to 1: for all $j$, we have $\sum_{i=1}^ka_{ij}=1$. A communication channel can be described by the set of transition matrices that it affords. Channel Q can be simulated by channel C if all transition matrices afforded by Q are convex combinations of transition matrices afforded by C. Such convex combinations occur naturally in information theory; they correspond to the sender and receiver having access to (unlimited) shared randomness. The relation `can be simulated by' is obviously reflexive and transitive. Two channels are equivalent if each can be simulated by the other.

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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[2] Michele Dall'Arno, "The signaling dimension of physical systems", Quantum Views 6, 66 (2022).

[3] Péter E. Frenkel and Mihály Weiner, "On entanglement assistance to a noiseless classical channel", Quantum 6, 662 (2022).

[4] Leevi Leppäjärvi, "Measurement simulability and incompatibility in quantum theory and other operational theories", arXiv:2106.03588, (2021).

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