Quantum Mechanical Operator Algebras and the logic of quantum measurement

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 mathematical structure called a 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. Evidently, this setup gives rise to a new empirical logical model of the quantum measurement problem. We note that all previous attempts to construct an empirical logical model for quantum mechanical measurement have failed. We conclude with a new generalization of the cubic lattice relating to higher spin systems, which leads us to new operator algebraic structures derived from its symmetry group.