  Explicitly separable density matrices are constructed for all separable
two-qubits states based on Hilbert-Schmidt (HS) decompositions. For density
matrices which include only two-qubits correlations the number of HS parameters
is reduced to 3 by using local rotations, and for two-qubits states which
include single qubit measurements, the number of parameters is reduced to 4 by
local Lorentz transformations. For both cases we related the absolute values of
the HS parameters to probabilities, and the outer products of various Pauli
matrices were transformed to pure states density matrices products. Simple
necessary and sufficient conditions for separability are derived. We discuss
related problems for three qubits. For n-qubits correlation systems the
sufficient condition for separability may be improved by local transformations,
related to high order singular value decompositions.

Quantum Physics