An Order-Theoretic Quantification of Contextuality

Abstract

In this essay, I develop order-theoretic notions of determinism and contextuality on domains and topoi. In the process, I develop a method for quantifying contextuality and show that the order-theoretic sense of contextuality is analogous to the sense embodied in the topos-theoretic statement of the Kochen-Specker theorem. Additionally, I argue that this leads to a relation between the entropy associated with measurements on quantum systems and the second law of thermodynamics. The idea that the second law has its origin in the ordering of quantum states and processes dates to at least 1958 and possibly earlier. The suggestion that the mechanism behind this relation is contextuality, is made here for the first time.

An Abstract Theory of Physical Measurements

“…The reification of this idea into the notion of measurement space seems to be sufficiently abstract that we do not end up bogged down in concrete details, but also, on the other hand, rich enough that something relevant is conveyed. In particular, despite not being a primitive building block of the theory, a notion of observer has emerged, and there are connections to observer (or context) dependent approaches [12,14,15,19,22,35,Ch. 12] which deserve being looked at.…”

An abstract theory of physical measurements

“…The reification of this idea into the notion of measurement space seems to be sufficiently abstract that we do not end up bogged down in concrete details, but also, on the other hand, rich enough that something relevant is conveyed. In particular, despite not being a primitive building block of the theory, a notion of observer has emerged, and connections to observer (or context) dependent approaches [8;9;13;17,Ch. 12;31;33] deserve being looked at.…”

Geometry of contextuality from Grothendieck’s coset space

“…The first dessin in Fig. 5b is of genus 0 and signature (B, W, F, g) = (21,35,9,0), and it stabilizes G H(2, 2) as shown in Fig. 5a through the stabilizer subgroup S 1 ∼ = Z 3 2 Z 2 2 .…”

“…In [20], a relative entropy of contextuality is defined. Finally in [21], a relation between the entropy associated with quantum measurements and the second law of thermodynamics is obtained in the frame of category theory. Our approach is different in that it is based on the shift between the quantum commutation/compatibility of operators and the commutation of corresponding coset representatives that serve as coordinates for these operators.…”

“…Once one accepts that the coset structure of finite geometries induced by dessins d'enfants may reveal a geometrical contextuality, it is tempting to arrive at a quantification of such a contextuality. Quantifying quantum contextuality is currently an active subject [12,19,20]. From our definition in Sec.…”