An Automated Deduction of the Independence of the Orthomodular Law from Ortholattice Theory

The optimization of quantum computing circuitry and compilers at some level must be expressed in terms of quantum-mechanical behaviors and operations. In much the same way that the structure of conventional propositional logic is the logic of the description of the behavior of classical physical systems and is isomorphic to a Boolean lattice, so also the algebra, C(H), of closed linear subspaces of (equivalently, the system of linear operators on) a Hilbert space is a logic of the descriptions of the behavior of quantum mechanical systems and is a model of an ortholattice (OL). An OL can thus be thought of as a kind of “quantum logic” (QL). C(H) is also a model of an orthomodular lattice (OML), which is an OL conjoined with an orthomodularity axiom/law (OMA). The rationalization of the OMA as a claim proper to physics has proven problematic, motivating the question of whether the OMA is required in an adequate characterization of QL. Here, I use an automated deduction framework to show that the OMA is independent of the axioms of ortholattice theory. These results corroborate (and fix a minor defect in) previously published work characterizing the strength of the OMA, and demonstrate the utility of automated deduction in investigating quantum computing logic-optimization strategies.