arXiv:1806.08227

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

Arkady Bolotin

Published Jun 19, 2018• 0 citations

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.

DOI: 10.48550/arxiv.1806.08227

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