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State spaces of orthomodular structures

Mirko Navara

Published Jan 1, 2000• 12 citations

Abstract

We present several known and one new description of orthomodular structures (orthomodular lattices, orthomodular posets and orthoalgebras). Originally, orthomodular structures were viewed as pasted families of Boolean algebras. Here we introduce semipasted families of Boolean algebras as an alternative description which is not as detailed, but substantially simplex. Semipasted families of Boolean algebras correspond to orthomodular structures in such a way that states and evaluation functionals are preserved. As semipasted families of Boolean algebras are quite general, they allow an easy construction of orthomodular structures with given state space properties. Based on this technique, we give a simplified proof of Shultz's Theorem on characterization of spaces of finitely additive states on orthomodular lattices. We also put some other results into the new context. We give a detailed exposition of the construction techniques as a tool for further applications, especially for finding counterexamples to questions about states on orthomodular structures.

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Sublattices and Δ-blocks of orthomodular posets

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Abstract States of quantum systems correspond to vectors in a Hilbert space and observations to closed subspaces. Hence, this logic corresponds to the algebra of closed subspaces of a Hilbert space. This can be considered as a complete lattice with orthocomplementation, but it is not distributive. It satisfies a weaker condition, the so-called orthomodularity. Later on, it was recognized that joins in this structure need not exist provided the subspaces are not orthogonal. Hence, the resulting structure need not be a lattice but a so-called orthomodular poset, more generally an orthoposet only. For orthoposets, we introduce a binary relation $\mathrel \Delta$ and a binary operator $d(x,y)$ that are generalizations of the binary relation $\textrm{C}$ and the commutator $c(x,y)$, respectively, known for orthomodular lattices. We characterize orthomodular posets among orthogonal posets. Moreover, we describe connections between the relations $\mathrel \Delta$ and $\leftrightarrow$ (the latter was introduced by P. Pták and S. Pulmannová) and the operator $d(x,y)$. In addition, we investigate certain orthomodular posets of subsets of a finite set. In particular, we describe maximal orthomodular sublattices and Boolean subalgebras of such orthomodular posets. Finally, we study properties of $\Delta$-blocks with respect to Boolean subalgebras and distributive subposets they include.

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Abstract This paper carries on the investigation of the orthomodular lattices that are endowed with a symmetric difference. Let us call them ODLs. Note that the ODLs may have a certain bearing on “quantum logics”-the ODLs are close to Boolean algebras though they capture the phenomenon of non-compatibility. The initial question in studying the state space of the ODLs is whether the state space can be poor. This question is of a purely combinatorial nature. In this note, we exhibit a finite ODL whose state space is empty (respectively, whose state space is a singleton).

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Reconstructing an atomic orthomodular lattice from the poset of its Boolean sublattices

Carmen Constantin, Doering, Andreas

We show that an atomic orthomodular lattice L can be reconstructed up to isomorphism from the poset B(L) of Boolean subalgebras of L. A motivation comes from quantum theory and the so-called topos approach, where one considers the poset of Boolean sublattices of L=P(H), the projection lattice of the algebra B(H) of bounded operators on Hilbert space.

arXiv:1306.1950•Jun 8, 2013•1 citation

We present several known and one new description of orthomodular structures (orthomodular lattices, orthomodular posets and orthoalgebras). Originally, orthomodular structures were viewed as pasted families of Boolean algebras. Here we introduce semipasted families of Boolean algebras as an alternative description which is not as detailed, but substantially simplex. Semipasted families of Boolean algebras correspond to orthomodular structures in such a way that states and evaluation functionals are preserved. As semipasted families of Boolean algebras are quite general, they allow an easy construction of orthomodular structures with given state space properties. Based on this technique, we give a simplified proof of Shultz's Theorem on characterization of spaces of finitely additive states on orthomodular lattices. We also put some other results into the new context. We give a detailed exposition of the construction techniques as a tool for further applications, especially for finding counterexamples to questions about states on orthomodular structures.

Abstract States of quantum systems correspond to vectors in a Hilbert space and observations to closed subspaces. Hence, this logic corresponds to the algebra of closed subspaces of a Hilbert space. This can be considered as a complete lattice with orthocomplementation, but it is not distributive. It satisfies a weaker condition, the so-called orthomodularity. Later on, it was recognized that joins in this structure need not exist provided the subspaces are not orthogonal. Hence, the resulting structure need not be a lattice but a so-called orthomodular poset, more generally an orthoposet only. For orthoposets, we introduce a binary relation

\mathrel \Delta

and a binary operator

d(x,y)

that are generalizations of the binary relation

\textrm{C}

and the commutator

c(x,y)

, respectively, known for orthomodular lattices. We characterize orthomodular posets among orthogonal posets. Moreover, we describe connections between the relations

\mathrel \Delta

and

\leftrightarrow

(the latter was introduced by P. Pták and S. Pulmannová) and the operator

d(x,y)

. In addition, we investigate certain orthomodular posets of subsets of a finite set. In particular, we describe maximal orthomodular sublattices and Boolean subalgebras of such orthomodular posets. Finally, we study properties of

\Delta

-blocks with respect to Boolean subalgebras and distributive subposets they include.

Sachs showed that a Boolean algebra is determined by its lattice of subalgebras. We establish the corresponding result for orthomodular lattices. We show that an orthomodular lattice L is determined by its lattice of subalgebras Sub(L), as well as by its poset of Boolean subalgebras BSub(L). The domain BSub(L) has recently found use in an approach to the foundations of quantum mechanics initiated by Butterfield and Isham, at least in the case where L is the orthomodular lattice of projections of a Hilbert space, or von Neumann algebra. The results here may add some additional perspective to this line of work.

Abstract This paper carries on the investigation of the orthomodular lattices that are endowed with a symmetric difference. Let us call them ODLs. Note that the ODLs may have a certain bearing on “quantum logics”-the ODLs are close to Boolean algebras though they capture the phenomenon of non-compatibility. The initial question in studying the state space of the ODLs is whether the state space can be poor. This question is of a purely combinatorial nature. In this note, we exhibit a finite ODL whose state space is empty (respectively, whose state space is a singleton).