  We study here the conditions to perform the distribution of a pure state on a
quantum network using quantum operations which can succeed with a non-zero
probability, the Stochastic Local Operation and Classical Communication (SLOCC)
operations.
  In their pioneering 2010 work, Kobayashi et al. showed how to convert any
classical network coding protocol into a quantum network coding protocol.
However, they left open whether the existence of a quantum network coding
protocol implied the existence of a classical one. Motivated by this question,
we characterize the set of distribution tasks achievable with non zero
probability for both classical and quantum networks. We develop a formalism
which encompasses both types of distribution protocols by reducing the solving
of a distribution task to the factorization of a tensor with complex
coefficients or real positive ones. Using this formalism, we examine the
equivalences and differences between both types of distribution protocols
exhibiting several elementary and fundamental relations between them as well as
concrete examples of both convergence and divergence. We answer by the negative
to the issue previously left open: some tasks are achievable in the quantum
setting, but not in the classical one. We believe this formalism to be a useful
tool for studying the extent of quantum network ability to perform multipartite
distribution tasks.
