We generalise some well-known graph parameters to operator systems by considering their underlying quantum channels. In particular, we introduce the quantum complexity as the dimension of the smallest co-domain Hilbert space a quantum channel requires to realise a given operator system as its non-commutative confusability graph. We describe quantum complexity as a generalised minimum semidefinite rank and, in the case of a graph operator system, as a quantum intersection number. The quantum complexity and a closely related quantum version of orthogonal rank turn out to be upper bounds for the Shannon zero-error capacity of a quantum channel, and we construct examples for which these bounds beat the best previously known general upper bound for the capacity of quantum channels, given by the quantum Lovasz theta number.
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