This paper formulates a notion of independence of subobjects of an object in a general (i.e., not necessarily concrete) category. Subobject independence is the categorial generalization of what is known as subsystem independence in the context of algebraic relativistic quantum field theory. The content of subobject independence formulated in this paper is morphism co-possibility: two subobjects of an object will be defined to be independent if any two morphisms on the two subobjects of an object are jointly implementable by a single morphism on the larger object. The paper investigates features of subobject independence in general, and subobject independence in the category of C*-algebras with respect to operations (completely positive unit preserving linear maps on C*-algebras) as morphisms is suggested as a natural subsystem independence axiom to express relativistic locality of the covariant functor in the categorial approach to quantum field theory.
This paper formulates a notion of independence of subobjects of an object in a general (i.e., not necessarily concrete) category. Subobject independence is the categorial generalization of what is known as subsystem independence in the context of algebraic relativistic quantum field theory. The content of subobject independence formulated in this paper is morphism co-possibility: two subobjects of an object will be defined to be independent if any two morphisms on the two subobjects of an object are jointly implementable by a single morphism on the larger object. The paper investigates features of subobject independence in general, and subobject independence in the category of C*-algebras with respect to operations (completely positive unit preserving linear maps on C*-algebras) as morphisms is suggested as a natural subsystem independence axiom to express relativistic locality of the covariant functor in the categorial approach to quantum field theory.
A review is made of the multitude of different mathematical formalizations of the physical concept ‘two observables (or two systems) are independent’ that have been proposed in quantum theories, particularly relativistic quantum field theory. The most basic mathematical formulation of independence in any quantum theory is what one may call kinematical independence: the two observables, resp. the observables of the two quantum systems, which are represented by elements of a C*-algebra, resp. two subalgebras of a C*-algebra, are required to commute. This is related to a mathematical formulation of the notion of the coexistence (or compatibility) of two observables. Another basic notion of independence, generally called statistical independence in the literature, is, roughly speaking, two quantum systems are said to be statistically independent if each can be prepared in any state, how ever the other system has been prepared. There are numerous mathematical formulations of this notion and their interrelationships are explained. Statistical independence and kinematical independence are shown to be logically independent. Additional notions such as strict locality and their relation to statistical independence are discussed. The mathematics of a more quantitative measure of statistical independence, Bell’s inequalities, is reviewed and its relations with previously introduced notions are indicated. All of these notions are then viewed in application to relativistic quantum field theory.
We propose a generalization of the description of Bell's inequalities in algebraic quantum field theory (AQFT) to the context of locally covariant quantum field theory (LCQFT). We use the functorial formulation of the state space as proposed in the seminal work \cite{BFV03} to show that for the suitable subcategory which consists of all possible admissible quadruples, yields a category of states over such observables which satisfy the Clauser-Horne-Shimony-Holt (CHSH) inequality. Such inequality is trivially preserved under algebraic morphisms and functoriality of the state space asserts us that it is preserved under isometric embeddings between the globally hyperbolic spacetimes where the algebras are defined and which constitute the locally covariant QFT functor.
There has been some recent interest in applying the techniques of Algebraic Quantum Field Theory (AQFT) to entanglement problems in perturbative QFT. In particular, the Hilbert space independence of this formulation makes it particularly interesting in the context of curved spacetimes and the emphasis on the algebra of observables makes the treatment of Bell inequalities in QFT resemble such treatment in non-relativistic Quantum Mechanics. In this work, we present the mathematical structures needed for formulating AQFT in terms of the Haag-Araki-Kastler (HAK) axioms and discuss their implications. Moreover, we discuss the algebraic approach to quantum entanglement in the form of Bell inequalities. We provide an extension of this formulation to general globally hyperbolic spacetimes using the so-called Locally Covariant approach to QFT, which extends the HAK axioms to general spacetimes by means of the Category Theory language.
This paper charts a very direct path between the categorical approach to quantum mechanics, due to Abramsky and Coecke, and the older convex-operational approach based on ordered vector spaces (recently reincarnated as "generalized probabilistic theories"). In the former, the objects of a symmetric monoidal category C are understood to represent physical systems and morphisms, physical processes. Elements of the monoid C(I,I) are interpreted somewhat metaphorically as probabilities. Any monoid homomorphism from the scalars of a symmetric monoidal category C gives rise to a covariant functor V_o from C to a category of dual-pairs of ordered vector spaces. Specifying a natural transformation u from V_o to 1 (where 1 is the trivial such functor) allows us to identify normalized states, and, thus, to regard the image category V_o(C) as consisting of concrete operational models. In this case, if A and B are objects in C, then V_o(A x B) defines a non-signaling composite of V_o(A) and V_o(B). Provided either that C satisfies a "local tomography" condition, or that C is compact closed, this defines a symmetric monoidal structure on the image category, and makes V_o a (strict) monoidal functor.
This paper proposes the Inter-inclusion Category (I-category) as a conceptual framework unifying the algebraic generation of quantum mechanics and the geometric constraints of general relativity. The central thesis is: generation and constraint are two projections of the same morphism. We define the I-category's basic structure—objects, morphisms, duals, projection conditions with three constraint conditions, Director, and structural functor—and demonstrate the projection condition through a minimal example. We then present the three-dimensional Kakeya conjecture (Wang-Zahl 2025) as a mathematical analog, showing its discrete-continuous isomorphism shares structural parallels with the I-category's projection conditions. Finally, we construct a discrete grid toy model with explicit mapping to I-category components, providing a protocol for verification. The rigorous construction of the Director remains open. We conclude with open problems and an invitation for collaboration.
What theoretical structures characterise our notions of what it means for a whole physical system to be the sum of its parts? We explore various aspects of this question, with emphasise on quantum theory, from the following perspective: A physical theory constitutes the definition of a set of propositions with logical relationships between them such that probabilities can be used to judge the merits of the theory, given empirical evidence. The logical relationships between the propositions characterise our semantic notions such as individuation. We distinguish between the ontological and epistemological notions of individuation. We introduce the notion of ontological individuation within the context of ontological models. A subsystem is defined as a proposition that provides necessary and sufficient information for making useful predictions. This definition to some extent generalises the notions of separability and local causality to define systems that may not be local. It highlights the unified motivations for many assumptions in the literature, such as measurement independence, and points to hidden assumptions of causality and mutual exclusivity that occur when defining systems. An epistemic no-signalling criterion gives rise to a tensor product structure for epistemic systems in quantum theory. We utilise a symmetry emergent from this structure to derive a bound on the observability of a large class of operations, dependent on only the purity of the quantum state. In particular, we derive an upper bound on the trace distance between an untransformed state and the state transformed by any trace preserving, unital operation. Our notions of epistemic systems also relate to distinguishability. It is shown that for measurements of any two (_nite outcome) particle quantum numbers on multi-fermion states, the total uncertainty, minimised over all states, is zero. This is because despite indistinguishability characterised as a constraint on the observable algebra, the appropriate correspondence between states of distinguishable and indistinguishable particles is one where the Hilbert space of indistinguishable particles relates to particles with an extra quantum number such that effective distinguishability can emerge. This Hilbert space is larger than the one for distinguishable particles and hence can (does) contain states for which the uncertainty relations for distinguishable particles are no longer valid. We conclude by arguing that in general, epistemic systems are not necessarily distinguishable; They may be only effectively distinguishable. This would imply a theory of indistinguishable modes. Several motivations are presented and the form of the theory suggested. It is hypothesised that such a theory may provide a relativistic quantum theory, alternative to quantum field theory, that satisfies cluster decomposition.