We study the $S$-matrix interpretation of quantum theory in light of Categotical Quantum Mechanics. The $S$-matrix interpretation of quantum theory is shown to be a functorial semantics relating the algebras of quantum theory to the effective $S$-matrix formalism. Consequently, issues such as state reduction and entanglement generation can be depicted in a simple manner. Moreover, this categorical $S$-matrix interpretation does not have the alleged thermodynamic cost.
We study the S-matrix interpretation of quantum theory in light of Categotical Quantum Mechanics. The S-matrix interpretation of quantum theory is shown to be a functorial semantics relating the algebras of quantum theory to the effective S-matrix formalism. Consequently, issues such as state reduction and entanglement generation can be depicted in a simple manner. Moreover, this categorical S-matrix interpretation does not have the alleged thermodynamic cost.
This paper explores the application of category theory to provide a semantic framework for quantum vector spaces, which are fundamental to quantum mechanics and quantum information. We move beyond the traditional algebraic and analytical descriptions of Hilbert spaces to adopt a more abstract, compositional, and graphical approach rooted in monoidal categories. This categorical perspective allows for a unified understanding of quantum phenomena, including superposition, entanglement, and measurement, by representing quantum systems as objects and processes as morphisms. We delve into the concept of compact closed categories and dagger symmetric monoidal categories as the appropriate mathematical structures to capture the essential features of quantum theory, such as duality, adjoints, and the tensor product for composite systems. Furthermore, the paper discusses how graphical calculi, arising from these categorical structures, offer an intuitive and rigorous method for reasoning about quantum protocols and computations, facilitating the design and verification of quantum algorithms. By providing a high-level, axiomatic framework, categorical semantics clarifies the foundational principles of quantum mechanics and opens new avenues for developing quantum programming languages and understanding the interplay between quantum information and computation.
In the present paper I describe a general formal semantic scheme for the interpretation of quantum mechanics (QM) , and on the basis of this scheme I examine the modal interpretation of QM — both the Copenhagen and the anti-Copenhagen variants — proposed by van Fraassen [19, 20, 21], This is intended to be a fragment of a larger work [12] which additionally investigates a number of closely related interpretations, including ones proposed by Bub and Demopoulos [1, 2, 3, 4], Fine [5, 6], and Krips [16, 17]. Formal semantically speaking (see, e.g., Thomason [18]), a logic L may be characterized as an ordered pair <SYN,SEM>, where SYN is the underlying syntax (language) of L, and SEM is the semantics of L, which consists of a class of semantic assignments on SYN.
Quantum mechanics is our most successful physical theory. However, it raises conceptual issues that have perplexed physicists and philosophers of science for decades. This 2004 book develops an approach, based on the proposal that quantum theory is not a complete, final theory, but is in fact an emergent phenomenon arising from a deeper level of dynamics. The dynamics at this deeper level are taken to be an extension of classical dynamics to non-commuting matrix variables, with cyclic permutation inside a trace used as the basic calculational tool. With plausible assumptions, quantum theory is shown to emerge as the statistical thermodynamics of this underlying theory, with the canonical commutation/anticommutation relations derived from a generalized equipartition theorem. Brownian motion corrections to this thermodynamics are argued to lead to state vector reduction and to the probabilistic interpretation of quantum theory, making contact with phenomenological proposals for stochastic modifications to Schrödinger dynamics.
This chapter, based on the background information of the previous chapter, will review the formalism of quantum mechanics – its fundamental mathematical structure (the basic contents of Hilbert space representation and operator algebra representation) and physical assumptions (kinematics, dynamics, and measurement theory). These are the premise and foundation for discussing the quantum histories interpretation. Then, in combination with the viewpoints of some literature in recent years, we will explore what is the interpretation of quantum mechanics, analyze the relationship between the theory of quantum mechanics and its interpretation, and declare the author’s basic stance on these two issues. These two are the primary issues that all interpretive work in the philosophy of physics will face (not limited to the interpretation of quantum mechanics), and the necessary clarification here is also conducive to the subsequent discussion. Three common terms – theory, formalism, and interpretation – are explicated and differentiated by offering the necessary definitions of their meanings, based on the works of Acuña and Ruetsche. The clarification will help to make the subsequent related discussions more precise.
QM∞ includes quantum field theory and the thermodynamic limit of quantum statistical mechanics — theories which, unlike the 'ordinary' quantum theories typically discussed by philosophers, concern infinite systems. The mathematical framework appropriate for the presentation of a theory of ordinary quantum mechanics is essentially unique. For theories of QM∞, this is not so. This prompts interpretive questions—for instance, about what makes a quantum theory the quantum theory it is — this work aims to chronicle. Having introduced a formal apparatus (operator algebra theory) suited to pursue these questions, the book articulates a variety of accounts of the content of quantum theories, accounts responsive to QM∞'s characteristic non-uniqueness. To evaluate these accounts, the book examines QM∞ settings (e.g. spontaneous symmetry breaking, cosmological particle creation, superconductivity) in which that characteristic non-uniqueness seems to matter, with a view toward determining which accounts sustain the uses to which the non-uniqueness is put. This approach not only brings work on the foundations of quantum theories in contact with the foundational investigation of other sorts of physical theories (e.g., thermodynamics, statistical mechanics, solid state physics, general relativity, and cosmology), it also brings the philosophy of physics in contact with other sorts of philosophy of science (e.g., accounts of explanation, reduction, and explanationist defenses of scientific realism). The book concludes that received notions of physical content and physical modality must be revised if they are to apply usefully to particular physical theories.
. A central theme in current work in quantum information and quantum foundations is to see quantum mechanics as occupying one point in a space of possible theories, and to use this perspective to understand the special features and properties which single it out, and the possibilities for alternative theories. Two formalisms which have been used in this context are operational theories , and categorical quantum mechanics . The aim of the present paper is to establish strong connections between these two formalisms. We show how models of categorical quantum mechanics have representations as operational theories. We then show how non-locality can be formulated at this level of generality, and study a number of examples from this point of view, including Hilbert spaces, sets and relations, and stochastic maps. The local, quantum, and no-signalling models are characterized in these terms. Introduction. A central theme in current work in quantum information and quantum foundations is to see quantum mechanics as occupying one point in a space of possible theories, and to use this perspective to understand the special features and properties which single it out, and the possibilities for alternative theories. Two formalisms which have been used in this context are operational theories [48, 41, 52, 47], and categorical quantum mechanics [6, 7]. • Operational theories allow general formulations of results in quantum foundations and quantum information [11, 12, 10]. They also play a prominent rôle in current work on axiomatizations of quantum mechanics [36, 19, 49, 25]. • Categorical quantum mechanics enables a high-level approach to quantum information and quantum foundations, which can be presented in terms of string-diagram representations of structures in monoidal categories [7]. This has proved very effective in providing a conceptually illuminating and technically powerful perspective on a range of topics, including quantum protocols [6], entanglement [24], measurement-based quantum computing [29], no-cloning [1], and non-locality [22]. The aim of the present paper is to establish strong connections between these two formalisms. We shall begin by reviewing operational theories. We then show how a proper formulation of compound systems within the operational framework leads to a view of operational theories as representations of monoidal categories of a particular form. We call these operational representations .