Varieties Of Quantum Measurement

Terence Love

This case study applies the Variety Dynamics (VD) analytical framework to foundational questions in physics and mathematics. It develops three related claims. First, varieties are informatic descriptions of reality — distinguishable options, states, decisions, or events — not the physical things they describe, and this distinction resolves persistent confusions in the foundations of physics including the quantum measurement problem. Second, quantum measurement, prediction, and physical theory are all informatic variety space operations; the wave function exhibits three modes of existence of which one — independent ontological status — is false; and the measurement problem dissolves when the informatic character of varieties is consistently maintained. Third, VD's variety space concept operates at the same foundational level as set theory and is arguably prior to it, since set theory's foundational operation of element distinguishability presupposes variety as an unnamed pre-formal primitive, and all specific physical state spaces (Hilbert space, phase space, Fock space) are special cases of variety space characterised by forgetful functors in the categorical sense. The case study also develops the connection between variety space and pre-sheaf structure in higher topos theory, where the variety-to-actualisation transition corresponds to sheafification. The analysis draws on the RaQM transaction cost argument established in Love (2026a) and provides evidence for Axiom 47's claim that VD operates at a more foundational level than physical theories. Axioms applied: 9, 25, 26, 32, 47, 48, 57 (in development).

Franck Laloë

Tobias J. Osborne

In this paper we study the problem of calculating the convex hull of certain affine algebraic varieties. As we explain, the motivation for considering this problem is that certain pure-state measures of quantum entanglement, which we call \emph{polynomial entanglement measures}, can be represented as affine algebraic varieties. We consider the evaluation of certain mixed-state extensions of these polynomial entanglement measures, namely \emph{convex and concave roofs}. We show that the evaluation of a roof-based mixed-state extension is equivalent to calculating a hyperplane which is multiply tangent to the variety in a number of places equal to the number of terms in an optimal decomposition for the measure. In this way we provide an \emph{implicit} representation of optimal decompositions for mixed-state entanglement measures based on the roof construction.

Geoff J. Pryde, Jeremy L. O’Brien, Andrew G. White +3 more

The use of entanglement allows a wide variety of quantum measurements outside the usual case of projective measurement. We describe several experimental photonic quantum measurement results relying on entangled measurement, such as weak measurements.