New Foundations for Quantum Information Theory

Logical information theory is the quantitative version of the logic of partitions just as logical probability theory is the quantitative version of the dual Boolean logic of subsets. The resulting notion of information is about distinctions, differences, and distinguishability, and is formalized as the distinctions of a partition (a pair of points distinguished by the partition). This paper is an introduction to the quantum version of logical information theory.

Quantum Logic of Direct-sum Decompositions

The usual quantum logic, beginning with Birkhoff and Von Neumann, was the logic of closed subspaces of a Hilbert space. This paper develops the more general logic of direct-sum decompositions of a vector space. This allows the treatment of measurement of any self-adjoint operators rather than just the projection operators associated with subspaces.

Logical Information Theory: New Foundations for Information Theory

There is a new theory of information based on logic. The definition of Shannon entropy as well as the notions on joint, conditional, and mutual entropy as defined by Shannon can all be derived by a uniform transformation from the corresponding formulas of logical information theory.

From Abstraction in Math to Superposition in QM

This is a draft paper that makes a perhaps surprising connection between the old Platonic notion of a paradigm-universal like ‘the white thing’ and an indefinite superposition state in quantum mechanics.

The Existence-Information Duality

The development of the logic of partitions (dual to the usual Boolean logic of subsets) and logical information theory bring out a fundamental duality between existence (e.g., elements of a subset) and information (e.g., distinctions of a partition). This leads in a more meta-physical vein to two different conceptions of reality, one of which provides the realistic interpretation of quantum mechanics.

Partition Logic talk slides Ljubljana

Slides from talk on Partition Logic at University of Ljubljana Sept. 8, 2015.

On the Objective Indefiniteness Interpretation of Quantum Mechanics

Classical physics and quantum physics suggest two different meta-physical conceptions of reality: the classical notion of a objectively definite reality “all the way down,” and the quantum conception of an objectively indefinite type of reality. Part of the problem of interpreting quantum mechanics (QM) is the problem of making sense out of an objectively indefinite reality. Our sense-making strategy is to follow the math by showing that the mathematical way to describe indefiniteness is by partitions (quotient sets or equivalence relations).

Partitions and Objective Indefiniteness in Quantum Mechanics

The problem of interpreting quantum mechanics (QM) is essentially the problem of making sense out of an objectively indefinite reality–that is described mathematically by partitions. Our sense-making strategy is implemented by developing the mathematics of partitions at the connected conceptual levels of sets and vector spaces. Set concepts are transported to (complex) vector spaces to yield the mathematical machinery of full QM, and the complex vector space concepts of full QM are transported to the set-like vector spaces over ℤ₂ to yield the rather fulsome pedagogical model of quantum mechanics over sets or QM/sets.

Talks and papers from 2013

Listing of talks and papers from 2013.

Introduction to Partition Logic

This is an introductory treatment of partition logic which also shows the extension to logical information theory and the possible killer application to quantum mechanics.