The assumption of the Hilbert lattice in the case of a two-dimensional system

TL;DR

This paper investigates the lattice structure of quantum propositions in two-dimensional systems, identifying assumptions that prevent the existence of a valuation map, contrasting with higher-dimensional cases.

Contribution

It provides an analysis of the Hilbert lattice assumptions in two-dimensional quantum systems, proposing modifications to exclude bivaluation maps.

Findings

Prime filters exist in the Hilbert lattice for qubits.

Certain lattice assumptions can be altered to prevent valuation maps in 2D.

The paper clarifies the logical structure of two-dimensional quantum systems.

Abstract

As it is known, the set of all closed linear subspaces of a Hilbert space together with a binary relation over the set represents the logic of the quantum propositions. It is also known that the lattices of the closed linear subspaces on a Hilbert space of dimension 3 or greater do not have a prime filter, hence those lattices do not allow a valuation map. In contrast to that, for qubits it is easy to find prime filters in the Hilbert lattice. This begs the question: What assumption(s) related to the lattices of the closed linear subspaces should be added or altered to preclude the bivaluation map in the two-dimensional case? The presented paper offers the answer to this question.