Completely depolarising channels, which transform every input state into white noise, are often regarded as the prototype of physical processes that are useless for communication. Here we show that the ability to combine N completely depolarising channels in a superposition of N alternative causal orders related by cyclic permutations enables a high-fidelity, heralded transmission of quantum information in the large N limit, and a deterministic transmission of quantum data assisted by classical communication for every N > 13. These phenomena highlight a fundamental difference with the N = 2 case, where the superposition of causal orders does not enable transmission of quantum data through completely depolarising channels. In addition, the ability to control N > 2 causal orders leads to an increase of the classical communication capacity, for which we derive an exact single-letter expression.
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