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Journal of Algebra and Its Applications

Jauch–Piron states on quantum logics

Michal Hroch, Pavel Pták

Published Dec 26, 2018• 0 citations
Publisher

Abstract

We show in this note that if [Formula: see text] is a Boolean subalgebra of the lattice quantum logic [Formula: see text], then each state on [Formula: see text] can be extended over [Formula: see text] as a Jauch–Piron state provided [Formula: see text] is Jauch–Piron unital with respect to [Formula: see text] (i.e. for each nonzero [Formula: see text], there is a Jauch–Piron state [Formula: see text] on [Formula: see text] such that [Formula: see text]). We then discuss this result for the case of [Formula: see text] being the Hilbert space logic [Formula: see text] and [Formula: see text] being a set-representable logic.

DOI
10.1142/s0219498820500176

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We show in this note that if [Formula: see text] is a Boolean subalgebra of the lattice quantum logic [Formula: see text], then each state on [Formula: see text] can be extended over [Formula: see text] as a Jauch–Piron state provided [Formula: see text] is Jauch–Piron unital with respect to [Formula: see text] (i.e. for each nonzero [Formula: see text], there is a Jauch–Piron state [Formula: see text] on [Formula: see text] such that [Formula: see text]). We then discuss this result for the case of [Formula: see text] being the Hilbert space logic [Formula: see text] and [Formula: see text] being a set-representable logic.
A quantum logic is the couple (L,M) where L is an orthomodular σ-lattice and M is a strong set of states on L. The Jauch–Piron property in the σ-form is also supposed for any state of M. A ‘‘tensor product’’ of quantum logics is defined. This definition is compared with the definition of a free orthodistributive product of orthomodular σ-lattices. The existence and uniqueness of the tensor product in special cases of Hilbert space quantum logics and one quantum and one classical logic are studied.
It has been shown by Birkhoff and v. Neumann (1936) and by Jauch and Piron (1963,1964,1968) that the subspaces of Hilbert space constitute an orthocomplemented quasi-modular lattice Lq, if one considers between two subspaces (elements) a, b the relation a⊆b and the operations a∩b, a∪b, a⊥. Furthermore, since the subspaces can be interpreted as quantum mechanical propositions, and since the operations ∩,∪ ,⊥ have some similarity with the logical operations ⋀ (and), ⋁ (or) and ⌝ (not), the question has been raised already by Birkhoff and v. Neumann, whether the lattice of subspaces of Hilbert space can be interpreted as a propositional calculus, sometimes called quantum logic. There are many kinds of lattices which can be interpreted as a propositional or logical calculus. A Boolean lattice LB of propositions corresponds to the calculus of classical logic and an implicative (Birkhoff, 1961) lattice Li has as a model the calculus of effective (intuitionistic) logic.
A Boolean orthoposet is the orthoposet P fulfilling the following condition: if a ^ b = 0 then a ? b. Boolean orthoposets enjoy some properties of Boolean algebras but, in some sense, they are far away of them (3, 4, 8). An important notion in quantum logic theory is the notion of Jauch-Piron state (2, 5, 6, 7). In this paper, we show connections between Boolean orthoposets and orthoposets with a proper number of Jauch-Piron states: We demonstrate that an orthoposet with enough two-valued Jauch-Piron states is Boolean and, on the other hand, we give several examples of Boolean orthoposets without any two- valued Jauch-Piron state.