A Generalization of Kochen-Specker Sets Relates Quantum Coloring to Entanglement-Assisted Channel Capacity

We introduce two generalizations of Kochen-Specker (KS) sets: projective KS sets and generalized KS sets. We then use projective KS sets to characterize all graphs for which the chromatic number is strictly larger than the quantum chromatic number. Here, the quantum chromatic number is defined via a nonlocal game based on graph coloring. We further show that from any graph with separation between these two quantities, one can construct a classical channel for which entanglement assistance increases the one-shot zero-error capacity. As an example, we exhibit a new family of classical channels with an exponential increase.Comment: 16 page

De Vries powers: a generalization of Boolean powers for compact Hausdorff spaces

We generalize the Boolean power construction to the setting of compact Hausdorff spaces. This is done by replacing Boolean algebras with de Vries algebras (complete Boolean algebras enriched with proximity) and Stone duality with de Vries duality. For a compact Hausdorff space $X$ and a totally ordered algebra $A$, we introduce the concept of a finitely valued normal function $f:X\to A$. We show that the operations of $A$ lift to the set $FN(X,A)$ of all finitely valued normal functions, and that there is a canonical proximity relation $\prec$ on $FN(X,A)$. This gives rise to the de Vries power construction, which when restricted to Stone spaces, yields the Boolean power construction. We prove that de Vries powers of a totally ordered integral domain $A$ are axiomatized as proximity Baer Specker $A$-algebras, those pairs $(S,\prec)$, where $S$ is a torsion-free $A$-algebra generated by its idempotents that is a Baer ring, and $\prec$ is a proximity relation on $S$. We introduce the category of proximity Baer Specker $A$-algebras and proximity morphisms between them, and prove that this category is dually equivalent to the category of compact Hausdorff spaces and continuous maps. This provides an analogue of de Vries duality for proximity Baer Specker $A$-algebras.Comment: 34 page

Two Forms of Inconsistency in Quantum Foundations

Recently, there has been some discussion of how Dutch Book arguments might be used to demonstrate the rational incoherence of certain hidden variable models of quantum theory (Feintzeig and Fletcher 2017). In this paper, we argue that the 'form of inconsistency' underlying this alleged irrationality is deeply and comprehensively related to the more familiar 'inconsistency' phenomenon of contextuality. Our main result is that the hierarchy of contextuality due to Abramsky and Brandenburger (2011) corresponds to a hierarchy of additivity/convexity-violations which yields formal Dutch Books of different strengths. We then use this result to provide a partial assessment of whether these formal Dutch Books can be interpreted normatively.Comment: 26 pages, 5 figure

A topos perspective on the Kochen-Specker theorem: II. Conceptual Aspects, and Classical Analogues:

In a previous paper, we have proposed assigning as the value of a physical quantity in quantum theory, a certain kind of set (a sieve) of quantities that are functions of the given quantity. The motivation was in part physical---such a valuation illuminates the Kochen-Specker theorem; and in part mathematical---the valuation arises naturally in the topos theory of presheaves. This paper discusses the conceptual aspects of this proposal. We also undertake two other tasks. First, we explain how the proposed valuations could arise much more generally than just in quantum physics; in particular, they arise as naturally in classical physics. Second, we give another motivation for such valuations (that applies equally to classical and quantum physics). This arises from applying to propositions about the values of physical quantities some general axioms governing partial truth for any kind of proposition.Comment: Small changes and correction

Extending Kolmogorov's axioms for a generalized probability theory on collections of contexts

Kolmogorov's axioms of probability theory are extended to conditional probabilities among distinct (and sometimes intertwining) contexts. Formally, this amounts to row stochastic matrices whose entries characterize the conditional probability to find some observable (postselection) in one context, given an observable (preselection) in another context. As the respective probabilities need not (but, depending on the physical/model realization, can) be of the Born rule type, this generalizes approaches to quantum probabilities by Auff\'eves and Grangier, which in turn are inspired by Gleason's theorem.Comment: 18 pages, 3 figures, greatly revise

Idempotent generated algebras and Boolean powers of commutative rings

A Boolean power S of a commutative ring R has the structure of a commutative R-algebra, and with respect to this structure, each element of S can be written uniquely as an R-linear combination of orthogonal idempotents so that the sum of the idempotents is 1 and their coefficients are distinct. In order to formalize this decomposition property, we introduce the concept of a Specker R-algebra, and we prove that the Boolean powers of R are up to isomorphism precisely the Specker R-algebras. We also show that these algebras are characterized in terms of a functorial construction having roots in the work of Bergman and Rota. When R is indecomposable, we prove that S is a Specker R-algebra iff S is a projective R-module, thus strengthening a theorem of Bergman, and when R is a domain, we show that S is a Specker R-algebra iff S is a torsion-free R-module. For an indecomposable R, we prove that the category of Specker R-algebras is equivalent to the category of Boolean algebras, and hence is dually equivalent to the category of Stone spaces. In addition, when R is a domain, we show that the category of Baer Specker R-algebras is equivalent to the category of complete Boolean algebras, and hence is dually equivalent to the category of extremally disconnected compact Hausdorff spaces. For a totally ordered R, we prove that there is a unique partial order on a Specker R-algebra S for which it is an f-algebra over R, and show that S is equivalent to the R-algebra of piecewise constant continuous functions from a Stone space X to R equipped with the interval topology.Comment: 18 page