Tensor Products of Quantum Mappings

In this paper, we examine properties of the tensor powers of quantum mappings $\Phi$. In particular, we review positivity properties of unitary and non-unitary qubit mappings $\Phi^{\otimes 2}$. For arbitrary finite-dimensional systems, we present the relationship between the positive and completely positive divisibility of dynamical mappings $\Phi_t^{\otimes 2}$ and $\Phi_t$. A criterion of annihilation of entanglement by an arbitrary qubit mapping $\Phi^{\otimes 2}$ is found.