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We extend the usual process-theoretic view on locality and causality in subsystems (based on the tensor product case) to general quantum systems (i.e.\ possibly non-factor, finite-dimensional von Neumann algebras). To do so, we introduce a…
We use non-standard analysis to define a category $^\star\!\operatorname{Hilb}$ suitable for categorical quantum mechanics in arbitrary separable Hilbert spaces, and we show that standard bounded operators can be suitably embedded in it. We…
We introduce a notion of global dimension for a triangulated category relative to a compact silting object. We prove that the finiteness of this dimension is an intrinsic property of the triangulated category itself and, therefore,…
We consider possible non-signaling composites of probabilistic models based on euclidean Jordan algebras. Subject to some reasonable constraints, we show that no such composite exists having the exceptional Jordan algebra as a direct…
This work discusses quantum states defined in a finite-dimensional Hilbert space. In particular, after the presentation of some of them and their basic properties the work concentrates on the group of the quantum optical models that can be…
In this chapter we survey some particular topics in category theory in a somewhat unconventional manner. Our main focus will be on monoidal categories, mostly symmetric ones, for which we propose a physical interpretation. These are…
We prove that for any finite-dimensional differential graded algebra with separable semisimple part the category of perfect modules is equivalent to a full subcategory of the category of perfect complexes on a smooth projective scheme with…
In this paper, I try once again to cause some good-natured trouble. The issue remains, when will we ever stop burdening the taxpayer with conferences devoted to the quantum foundations? The suspicion is expressed that no end will be in…
An understanding of quantum theory in terms of new, underlying descriptions capable of explaining the existence of non-classical correlations, non-commutativity of measurements and other unique and counter-intuitive phenomena remains still…
We consider categorical logic on the category of Hilbert spaces. More generally, in fact, any pre-Hilbert category suffices. We characterise closed subobjects, and prove that they form orthomodular lattices. This shows that quantum logic is…
In this paper, we present a comprehensive system for the treatment of the topic of limits--conceptually, computationally, and formally. The system addresses fundamental linguistic flaws in the standard presentation of limits, which attempts…
We define generalized bialgebras and Hopf algebras and on this basis we introduce quantum categories and quantum groupoids. The quantization of the category of linear (super)spaces is constructed. We establish a criterion for the classical…
In fairly elementary terms this paper presents, and expands upon, a recent result by Garner by which the notion of topologicity of a concrete functor is subsumed under the concept of total cocompleteness of enriched category theory.…
We introduce a hierarchy of linear systems for showing that a given subspace of pure quantum states is entangled (i.e., contains no product states). This hierarchy outperforms known methods already at the first level, and it is complete in…
Higher category theory is an exceedingly active area of research, whose rapid growth has been driven by its penetration into a diverse range of scientific fields. Its influence extends through key mathematical disciplines, notably homotopy…
To any dg-category $T$ (over some base ring $k$), we define a $D^{-}$-stack $\mathcal{M}_{T}$ in the sense of \cite{hagII}, classifying certain $T^{op}$-dg-modules. When $T$ is saturated, $\mathcal{M}_{T}$ classifies compact objects in the…
The exponential modalities of linear logic have been used by various authors to model infinite-dimensional quantum systems. This paper explains how these modalities can also give rise to the complementarity principle of quantum mechanics.…
Several categories look like categories of relations, but do not fit the established theory of relations in regular categories. They include the category of surjective multivalued functions, the category of injective partial functions, the…
The rather unintuitive nature of quantum theory has led numerous people to develop sets of (physically motivated) principles that can be used to derive quantum mechanics from the ground up, in order to better understand where the structure…
Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of…