  In quantum mechanics, the connection between the operator algebraic
realization and the logical models of measurement of state observables has long
been an open question. In the approach that is presented here, we introduce a
new application of the cubic lattice. We claim that the cubic lattice may be
faithfully realized as a subset of the self-adjoint space of a von Neumann
algebra. Furthermore, we obtain a unitary representation of the symmetry group
of the cubic lattice. In so doing, we re-derive the classic quantum gates and
gain a description of how they govern a system of qubits of arbitrary
cardinality.
