Quantization by Internalization

Quantum mechanics is the fundamental theory in physics that revolutionized our understanding of the behavior of nature at the smallest scales. Quantum mechanics introduces several concepts that challenge our everyday intuition such as the superposition of states, the uncertainty principle and entanglement. Many of these concepts can be understood better from the mathematical formalism underlying quantum mechanics. An important ingredient of this formalism is the idea of describing observables of quantum systems by (possibly infinite) matrices rather than by number-valued functions as in the case of classical systems. In contrast to these functions, matrices do not always commute: the order of multiplication matters. Several counter-intuitive aspects of quantum physics, such as the uncertainty principle, can be understood from the noncommutative mathematical framework of quantum physics. To find a mathematical model for some quantum phenomenon, one typically identifies a classical counterpart of this phenomenon, and somehow replaces functions in the mathematical description of this counterpart by matrices in order to obtain the required model for the quantum phenomenon. This step from functions to matrices is called quantization. Unfortunately, a priori, it is not clear with what matrix one should replace a given function in a classical model. This has led to several quantization methods, which give procedures for choosing this matrix. Nevertheless, there are still several mathematical structures to which the current quantization methods do not apply, but which are used for the description of classical counterparts of known quantum phenomena, in particular in quantum computing and quantum information theory. A reason for this is that many mathematical structures are described in terms of relations instead of functions, and most quantization methods can only quantize functions. However, relatively recently, Kuperberg and Weaver discovered how to quantize relations, resulting in the notion of a quantum relation. In this proposal, we aim to study how to quantize mathematical structures using these quantum relations in a systematic and consistent way via category theory, which is a mathematical theory in which mathematical structures are studied and represented in a category consisting of a collection of objects representing a mathematical structure and a collection of morphisms, which are functions or relations that preserve the structure. Examples include the categories Rel of sets and relations, and qRel of quantum sets and quantum relations. An important technique in category theory is internalization, the generalization of set-theoretic constructions to other categories. Since qRel shares many properties with Rel, most structures can be internalized in qRel, and the resulting internalized structures are precisely the quantized versions of these structures. Hence, we intend to study quantization via this internalization process.