Complete Factorization of Entangled Quantum States PART 2: Factorizing the GHZ_n and W States Using Semi-Structured Complex Numbers

This paper is a second in a three-part series providing an auxiliary parameter method for factorizing maximally entangled states. A major vulnerability of entangled states is they cannot be factored, leaving the whole composite system highly susceptible to decoherence. Factorizing these states creates a protective barrier that prevents local noise or measurements in one subsystem corrupting the rest of the system. This paper introduces an auxiliary parameter method that uses semi-structured complex numbers to create (1) an unphysical auxiliary parameter k_n and (2) a unique j-conjugate framework to embed maximally entangled systems in semi-structured state space and factorize them into independent single-qubit product states. To demonstrate this method the maximally entangled GHZ_n and W States were completely factorized. These factorizations strictly satisfy classical local realism, passing the Mermin Inequality Test (for GHZ-type states) and the MARS inequality test (for W states) whilst in semi-structured state space. Finally, an inverse transformation maps these auxiliary terms back to zero, recovering the original standard Euclidean quantum states. This framework provides a novel algebraic tool for manipulating maximally entangled states as formal tensor product states.