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Quantum Computation and Quantum Information are continuously growing researchareas which are based on nature and resources of quantum mechanics, assuperposition and entanglement. In its gate array version, the use ofconvenient and appropriate gates is essential. But while those proposed gatesadopt convenient forms for computational algorithms, in the practice, theirdesign depends on specific quantum systems and stuff being used. Gates designis restricted to properties and limitations of interactions and physicalelements being involved, where Quantum Control plays a deep role. Quantumcomplexity of multipartite systems and their interactions requires a tightcontrol to manipulate their quantum states, either local and non-local ones,but still a reducibility procedure should be addressed. This work shows how ageneral $2d$-partite two level spin system in $SU(2d)$ could be decomposed in$2^{n-1}$ subsystems on $SU(2)$, letting establish control operations. Inparticular, it is shown that Bell gems basis is a set of natural states onwhich decomposition happen naturally under some interaction restrictions. Thus,alternating the direction of local interaction terms in the Hamiltonian, thisprocedure states a universal exchange semantics on those basis. The structuredeveloped could be understood as a splitting of the $2d$ information channelsinto $2^{2d-1}$ pairs of $2$ level information subsystems.

Bell gems naturally split dynamics information from $SU(2^{2d}) \rightarrow U(1)^{2^{2d-1}-1} \times SU(2)^{2^{2d-1}}$