Checking strict positivity of Kraus maps is NP-hard

Basic properties in Perron–Frobenius theory are positivity, primitivity, and irreducibility. Whereas these properties can be checked in polynomial time for stochastic matrices, we show that for Kraus maps – the noncommutative generalization of stochastic matrices – checking positivity is NP-hard. This is in contrast with irreducibility and primitivity, which we show to be checkable in strongly polynomial time for completely positive maps – the noncommutative generalization of nonpositive matrices. As an intermediate result, we get that the bilinear feasibility problem over Q is NP-hard.