  Exploiting the cone structure of the set of unnormalized mixed quantum
states, we offer an approach to detect separability independently of the
dimensions of the subsystems. We show that any mixed quantum state can be
decomposed as $\rho=(1-\lambda)C_{\rho}+\lambda E_{\rho}$, where $C_{\rho}$ is
a separable matrix whose rank equals that of $\rho$ and the rank of $E_{\rho}$
is strictly lower than that of $\rho$. With the simple choice
$C_{\rho}=M_{1}\otimes M_{2}$ we have a necessary condition of separability in
terms of $\lambda$, which is also sufficient if the rank of $E_{\rho}$ equals
1. We give a first extension of this result to detect genuine entanglement in
multipartite states and show a natural connection between the multipartite
separability problem and the classification of pure states under stochastic
local operations and classical communication (SLOCC). We argue that this
approach is not exhausted with the first simple choices included herein.
