  Quantum entanglement has been regarded as one of the key physical resources
in quantum information sciences. However, the determination of whether a mixed
state is entangled or not is generally a hard issue, even for the bipartite
system. In this work we propose an operational necessary and sufficient
criterion for the separability of an arbitrary bipartite mixed state, by virtue
of the multiplicative Horn's problem. The work follows the work initiated by
Horodecki {\it et. al.} and uses the Bloch vector representation introduced to
the separability problem by J. De Vicente. In our criterion, a complete and
finite set of inequalities to determine the separability of compound system is
obtained, which may be viewed as trade-off relations between the quantumness of
subsystems. We apply the obtained result to explicit examples, e.g. the
separable decomposition of arbitrary dimension Werner state and isotropic
state.
