Explicit constructions of all separable two-qubits density matrices and related problems for three-qubits systems

Abstract: 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.