The problem of computing the quantum dynamical entropy introduced by Alicki and Fannes requires the trace of the operator function $\Omega \log \Omega$, where $\Omega$ is a non-negative, Hermitean operator. In the numerical analysis of this problem, $\Omega$ must be reduced to a matrix of large order. We study its properties and we derive efficient algorithms to solve this problem, implementable on parallel machines with distributed memory.