We first present a general theory of diagrams, and in particular, of string diagrams, and discuss why diagrams are a very natural starting point for developing scientific theories. [1] The category FdHilb consists of a symmetric monoidal category (SMC) with finite-dimensional complex Hilbert spaces as objects and linear transformations as arrows. 415–425, IEEE Comput. Mathematically, quantum mechanics can be regarded as a non-classical probability calculus resting upon a non-classical propositional logic. This is the common representation for an empty diagram in categorical quantum mechanics. We show the existence of unital special commutative $\dagger$-Frobenius algebras, and we conclude … • In Section 1.3, we review the categorical approach to quantum mechanics. In particular, we demonstrate that the categorical structures implied by the polymorphically typed connectives give rise to a (lax unitless) form of the special forms of Frobenius algebras known as classical structures, used heavily in abstract categorical approaches to quantum mechanics. I Why complex numbers? Then we … Captions. Categorical Quantum Mechanics gives rise to more than just graphical calculi. Moreover, the sender may not know the … Abstract. 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. see today, in quantum mechanics there is an even more direct connection between the conserved quantities and the symmetries: the operator for the total conserved quan-tity (i.e. categorical approach will have to be modified to accommodate infinite dimensional quantum mechanics. The long-term goal of our research is to develop a powerful quantum logic which is useful in the formal verification of quantum programs and protocols. I Study of compositional nature of (physical) systems Primitive notion: forming compound systems I Operational yet algebraic I Why non-unit state vectors? Quantum teleportation is a technique for transferring quantum information from a sender at one location to a receiver some distance away. It uses compound systems as the primitive notion, which is formalised by using symmetric monoidal categories. Taking a more general categorical standpoint of which the above notion is an instance, in Section 3 we advance the view that an object of inquiry in a category proof theory and no-go theorems in quantum mechanics. Yet again, Categorical quantum mechanics offers much more than just a rigorous framework for graphical calculi. quantum gravity • clarify the relationship between quantum and classical computation • suggest new primitives for the design and analysis of a generalization of the methods of categorical quantum mechanics [5] to the case where the systems are of arbitrary dimensions. More recently, I have been working on a unified sheaf-theoretic approach to non-locality and contextuality. Title: Categorical quantum mechanics II: Classical-quantum interaction: Author(s): Coecke, B.; Kissinger, A. For instance, in matrix 1/30 Motivation Mixed Unitary Categories Unitary construction Dagger compact closed categories This work is available at arXiv:1809.00275 Dagger compact closed categories (y-KCC) provide a categorical This formalism admits a purely graphical calculus, but the causal structure of these diagrams, and the formalism in general, is unclear. Categorical QM aims to reconstruct quantum mechanics in terms of minimal algebraic structures: • discover the mathematical heart of the theory • suggest constraints on successor theories e.g. The same relational algebra generates a number of group structures, among them W ∞. Categorical quantum mechanics (CQM) uses symmetric monoidal categories to formalize quantum theory, in order to extract the key structures that yield protocols such as teleportation in an abstract way. We introduce a realization of the corresponding categorical algebra of compositions, which naturally gives rise to the fundamental quantum laws, thus indicating category theory as the foundation of quantum mechanics. Title: Categorical Quantum Mechanics II: Classical-Quantum Interaction. Compact closed categories, Frobenius and Bi algebras are also the building blocks of the categorical approach to Quantum Mechanics, known under the acronym CQM. It is a thoroughly principled framework that sheds light on the underlying compositionality of the physical world. In this paper we introduce the basic idea of our categorical logic of quantum programs (CLQP): It combines the logic of quantum programming (LQP) and categorical quantum mechanics (CQM) such that the advantages … We derive the category-theoretic backbone of quantum theory from a process ontology. Abramsky and Coecke (Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, pp. This is the second part, the first part being [13], of a three-part overview on categorical quantum mechanics (CQM), an area of applied category-theory … More specifically, in quantum mechanics each probability-bearing proposition of the form “the value of physical quantity \(A\) lies in the range \(B\)” is represented by a projection operator on a Hilbert space \(\mathbf{H}\). This leads to an elegant formalism for describing quantum protocols such as quantum teleportation. Applications of non-standard analysis to quantum theory already appeared in the past decades [6,8,10], mainstream quantum mechanics which the categorical framework has thus failed to adequately capture, and we demonstrate how they can be used to recover a great deal of CQM machinery in infinite-dimensions. In this paper we present a new categorical approach which attempts to provide an original understanding of QM. We begin by considering the signaling of quantum contextuality, the endo-theoretic notion of perspective in the case of Hilbert-space quantum mechanics. This realization of quantum logic within the categorical approach is based on the simple notion of direct product decompositions, and may be sufficiently resilient to persist through, and help guide, such modifications. categorical quantum mechanics Robin Cockett, Cole Comfort, and Priyaa Srinivasan University of Calgary 0/30. I Why non-hermitean operators? This leads to an elegant formalism for describing quantum protocols such as quantum teleportation. A link between quantum logic and categorical quantum mechanics. Categorical quantum mechanics is a way of formalising the structural features of quantum theory using category theory. Publication year: 2016 Another connection to Quantum formalisms, is the structure preserving passage from grammatical structure to vectorial meaning, which is through a functor similar to the Quantization functor of Topological Quantum … Date: 15 May 2019: Source: Own work: Author: johnie102: Licensing . More specifically, we treat quantum theory as a theory of systems, processes, and their interactions. the charge) is the same operator that gives the change in the state under an inflnitesimal version of the associated symmetry. While teleportation is commonly portrayed in science fiction as a means to transfer physical objects from one location to the next, quantum teleportation only transfers quantum information. Categorical quantum mechanics was conceived by Abramsky and Coecke about ten years ago in an attempt to clarify the conceptual foundations of the subject, while TQFT (pioneered by Atiyah, Segal and Witten) is one of the finest modern examples of the interaction between pure mathematics and theoretical physics. Complementarity in categorical quantum mechanics Chris Heunen September 10, 2010 Abstract We relate notions of complementarity in three layers of quantum me-chanics: (i) von Neumann algebras, (ii) Hilbert spaces, and (iii) ortho-modular lattices. Categorical quantum computation Mark Richard Przepiora December 20, 2009 1 Introductionandmotivation DiracpublishedThePrinciplesofQuantumMechanics in1930,inwhichhe It uses compound systems as the primitive notion, which is formalised by using symmetric monoidal categories. This generalization is a logical next step to extend the existing categorical framework, and is one of practical importance. We show how models of categorical quantum mechanics have representations as operational theories. Description: English: This is the common representation for an empty diagram in categorical quantum mechanics. Categorical Quantum Mechanics, and graphical calculi in particular, has proven to be an intuitive and powerful way to reason about quantum computing. Download PDF: Sorry, we are unable to provide the full text but you may find it at the following location(s): http://arxiv.org/pdf/0808.1023 (external link) Categorical quantum mechanics is a way of formalising the structural features of quantum theory using category theory. I pioneered categorical quantum mechanics with Bob Coecke. Summary . The further contents of the paper are as follows: • In the next section, we briefly review the three-way link between logic, computation, and categories and recall Joyal’s lemma. Our logos categorical approach attempts to consider the main features of the quantum formalism as the standpoint to develop a conceptual representation that explains what the theory is really talking about —rather than as problems that need to be bypassed in … I Powerful graphical calculus I Allows different interpretation in many different fields I Physics: quantum theory, quantum … Soc., New York, 2004) have recently introduced an approach to finite dimensional quantum mechanics based on strongly compact closed categories with biproducts.In this note it is shown that the projections of any object A in such a … arXiv:quant-ph/9906123. Let us now apply the concept of a symmetric monoidal category (SMC) to quantum mechanics in finite dimensions.In fact, the first step is rather straight-forward: Definition 1. Category theory and quantum mechanics Posted on January 21, 2020 by Peter Smith My last link to something categorical turned out to be pointing to a less-than-splended online resource. This course is taught by the Department of Computer Science, for further information, please click on: http://www.cs.ox.ac.uk/teaching/courses/2018-2019/cqm/ Arrow composition is provided by … Taking a more general categorical perspective of which Categorical quantum mechanics is the study of quantum computation and quantum foundations using category theory. In particular, categorical quantum mechanics … International Journal of Theoretical Physics , 48 (3), 769–802. Categorical Quantum Mechanics? We then show how nonlocality 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 ZX-calculus, a graphical language of quantum computation developed in the framework of CQM, is introduced by Coecke and Duncan [ 24 , 27 ]. Hardy , L. N.d. Disentangling nonlocality and teleportation.
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