Authors: Davide Fazio and Raffaele Mascella

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Our talk concerns an order-theoretical and algebraic investigation of some partially ordered algebraic structures arising in the logico-algebraic approach to Quantum Mechanics (QM), namely paraorthomodular lattices (PLs). Such algebras find a prominent example in the boundedly complete pseudo-Kleene Olson-de Groote spectral lattice of effects over a separable Hilbert space \(\mathcal{H}\).

It is well known that the set \(\mathbf{P}(\mathcal{H})\) of projections over a separable Hilbert space (HS) \(\mathcal{H}\) can be regarded as quantum experimental propositions concerning Lüders measurements which form a complete orthomodular lattice (OML). OMLs provide a logico-algebraic rendering of complementarity phenomena between observables, as they can be regarded as “pastings” of their maximal Boolean “contexts”/sub-algebras (blocks). This means that the order of an OML determines, and it is determined by, the order on its Boolean sub-algebras.

Over the past years, it has been argued that the idealizing and simplifying assumption of considering only Lüders measurements leads to conflict with experimental possibilities and to theoretical inconsistencies. As a consequence, generalized measurable “unsharp” properties, whose formal counterpart is given by effects, have been introduced. In order to provide them with a logical structure which is amenable of lattice-theoretical investigations, PLs, i.e. “unsharp” generalizations of OMLs, have been defined.

In the first part of our talk we characterize the class of paraorthomodular lattices which are “pastings” of their Kleene (i.e. distributive) blocks. To this aim we introduce the variety of super-paraorthomodular lattices (SPLs), i.e. pseudo-Kleene lattices satisfying a generalized version of the orthomodular law [1]. We will outline their salient order-theoretical and algebraic properties by providing several characterization results like e.g. a Dedekind-type (forbidden configuration) theorem, and a theory of commutativity (for modular SPLs). It will turn out that the spectral pseudo-Kleene lattice \(\mathbf{L}(\mathcal{H})\) of effects over a separable HS \(\mathcal{H}\) is indeed super-paraorthomodular. Moreover, SPLs’ theory captures important features of spectral lattices as, for example, orthocomplemented elements of an SPL \(\mathbf{A}\) always form an orthomodular poset.

In the second part of the talk, taking advantage of well known J. Czelakowski’s results, we introduce paraconsistent partial referential matrices (PPRM) for the formal treatment of (partial) experimental quantum propositions taking truth values over the three element Kleene chain \(\mathbf{K}_{3}\), which are meaningful/defined for certain states of a physical system, and undefined otherwise. We will show that any super-praorthomodular lattice is isomorphic to a pseudo-Kleene lattice of partial propositions in the algebraic reduct of a PPRM.

Bibliography

1. D. Fazio, and R. Mascella, On Contextuality and Unsharp Quantum Logic,submitted, arXiv:2311.06109.