[Paper Review] Further remarks on an order for quantum observables

Jānis Cı̄rulis|arXiv (Cornell University)|Jan 3, 2013

Advanced Algebra and Logic4 references3 citations

TL;DR

This paper extends the logical order for bounded self-adjoint operators on a Hilbert space by introducing a total skew meet operation and proving that the structure forms a right normal skew nearlattice. It establishes that the logical order induces a generalized orthoalgebra with orthomodular initial segments, and provides explicit operator-theoretic descriptions of Gudder's join and meet using projections and composition, offering a new algebraic framework for quantum observables beyond standard lattices.

ABSTRACT

S. Gudder and, later, S. Pulmanova and E. Vincekova, have studied in two recent papers a certain ordering of bounded self-adjoint operators on a Hilbert space. We present some further results on this ordering and show that some structure theorems of the ordered set of operators can be obtained in a more abstract setting of posets having the upper bound property and equipped with a certain orthogonality relation.

Motivation & Objective

• To extend the logical order on bounded self-adjoint operators by introducing a total skew meet operation.
• To show that the poset of quantum observables under this order forms a right normal skew nearlattice.
• To provide explicit operator-theoretic expressions for Gudder's join and meet using projections and composition.
• To establish that every initial segment of the logical order is an orthomodular lattice, generalizing earlier results.
• To unify quantum logic structures with information systems via abstract poset-theoretic properties like the upper bound property and orthogonality.

Proposed method

• The paper uses abstract poset theory, focusing on nearsemilattices with the upper bound property and a defined orthogonality relation.
• It introduces the skew meet operation as a total binary operation on the set of bounded self-adjoint operators, defined via the maximal commuting projection within the range of the operators.
• The construction relies on the existence of a greatest element in the set of projections commuting with both operands and contained in the range of the first operator.
• The paper proves that the skew meet operation is associative and idempotent, and satisfies specific order-theoretic identities involving the logical order and orthogonality.
• It applies known structure theorems from semigroup and lattice theory to show that the algebraic system forms a right normal skew nearlattice.
• The proof of bounded completeness is established through the existence of a maximal commuting projection within the range of the operators, ensuring the existence of joins and meets.

Experimental results

Research questions

• RQ1 Can the logical order on bounded self-adjoint operators be characterized algebraically using a total binary operation like the skew meet?
• RQ2 Does every initial segment of the logical order on quantum observables form an orthomodular lattice?
• RQ3 What is the precise operator-theoretic expression for the Gudder join and meet in terms of projections and composition?
• RQ4 How do the properties of the logical order relate to abstract poset structures such as nearsemilattices and quasi-orthomodular lattices?
• RQ5 Can the skew nearlattice structure unify quantum logic with information systems theory via shared algebraic axioms?

Key findings

• The skew meet operation on the set of bounded self-adjoint operators is total and associative, forming an idempotent semigroup with specific order properties.
• The logical order induces a generalized orthoalgebra structure on the set of quantum observables, with every initial segment being an orthomodular lattice.
• The Gudder join and meet can be explicitly described using the composition of operators and lattice operations on projections, particularly via the maximal commuting projection in the range.
• The poset of bounded self-adjoint operators under the logical order is bounded complete, meaning every subset bounded above has a supremum.
• The structure of the logical order is shown to be a right normal skew nearlattice, linking quantum logic with information systems theory through shared algebraic axioms.
• The paper provides a simple proof of bounded completeness by constructing the supremum via the maximal projection commuting with both operators and contained in the range of the first.