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

TL;DR

This paper presents explicit constructions of all separable two-qubit density matrices using Hilbert-Schmidt decompositions, simplifies parameter counts with local transformations, and discusses extensions to three-qubit systems.

Contribution

It provides explicit methods for constructing all separable two-qubit states and introduces parameter reduction techniques using local transformations, with implications for three-qubit systems.

Findings

Explicit constructions for all separable two-qubit states.

Reduced parameter counts via local rotations and Lorentz transformations.

Derived simple necessary and sufficient conditions for separability.

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.