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AIP conference proceedings

Kindergarten Quantum Mechanics: Lecture Notes

Bob Coecke

Published Jan 1, 2006• 74 citations
Publisher

Abstract

These lecture notes survey some joint work with Samson Abramsky as it was presented by me at several conferences in the summer of 2005. It concerns ‘doing quantum mechanics using only pictures of lines, squares, triangles and diamonds’. This picture calculus can be seen as a very substantial extension of Dirac’s notation, and has a purely algebraic counterpart in terms of so‐called Strongly Compact Closed Categories (introduced by Abramsky and I in [3, 4]) which subsumes my Logic of Entanglement [11]. For a survey on the ‘what’, the ‘why’ and the ‘hows’ I refer to a previous set of lecture notes [12, 13]. In a last section we provide some pointers to the body of technical literature on the subject.

DOI
10.1063/1.2158713

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These lecture notes survey some joint work with Samson Abramsky as it was presented by me at several conferences in the summer of 2005. It concerns ‘doing quantum mechanics using only pictures of lines, squares, triangles and diamonds’. This picture calculus can be seen as a very substantial extension of Dirac’s notation, and has a purely algebraic counterpart in terms of so‐called Strongly Compact Closed Categories (introduced by Abramsky and I in [3, 4]) which subsumes my Logic of Entanglement [11]. For a survey on the ‘what’, the ‘why’ and the ‘hows’ I refer to a previous set of lecture notes [12, 13]. In a last section we provide some pointers to the body of technical literature on the subject.
These lecture notes survey some joint work with Samson Abramsky as it was presented by me at several conferences in the summer of 2005. It concerns `doing quantum mechanics using only pictures of lines, squares, triangles and diamonds'. This picture calculus can be seen as a very substantial extension of Dirac's notation, and has a purely algebraic counterpart in terms of so-called Strongly Compact Closed Categories (introduced by Abramsky and I in quant-ph/0402130 and [4]) which subsumes my Logic of Entanglement quant-ph/0402014. For a survey on the `what', the `why' and the `hows' I refer to a previous set of lecture notes quant-ph/0506132. In a last section we provide some pointers to the body of technical literature on the subject.
These lecture notes survey some joint work with Samson Abramsky as it was presented by me at Mathematical Foundations of Programming Semantics XXI in Birmingham (2005).2 It concerns a categorical semantics for quantum mechanics introduced in [Abramsky, S. and Coecke, B. (2004) A categorical semantics of quantum protocols. In: Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS'04), IEEE Computer Science Press. An extended & improved version is available at http://arXiv.org/quant-ph/0402130; Abramsky, S. and Coecke, B. (2005) Abstract physical traces. Theory and Applications of Categories 14, 111–124. Available at www.tac.mta.ca/tac/volumes/14/6/14-06abs.html]. We present this semantics with a particular focus on the connection between categorical diagrams and certain pictures involving typed squares, triangles, diamonds and lines. Along the way we unravel the structural components which come with strong compact closure. We provide pointers to related literature. Keywords: Strong compact closure, semantics, quantum mechanics, quantum logic, quantum informatics.
There is a huge number of excellent and comprehensive textbooks on quantum mechanics. They mainly differ for the approach, more or less oriented to the formalism rather than to the phenomenology, as well as for the topics covered. These lectures have been based mainly on the classical textbook by Gasiorowicz (1974). I must confess that the main reason for my choice of Gasiorowicz (1974) is affective, as it was the textbook where I first learned the basic principles of quantum mechanics. Beyond my personal taste, I now recognize that Gasiorowicz (1974) is still a very good textbook on quantum mechanics, with a rigorous theoretical approach accompanied by a wide collection of applications. If the textbook by Gasiorowicz was my main basis, I have taken much also from other textbooks such as Phillips (2003), as well as from the excellent classical textbook by Dirac (1981). In order to avoid complications in the mathematics and in the notation, the topic is presented in these notes with reference to one-dimensional systems, with just a few marginal extensions to the three-dimensional formulation.
This paper surveys joint work with Samson Abramsky. I will somewhat informally discuss the main results of a series of papers in a pedestrian not too technical way. These include: . 'The logic of entanglement', that is, the identification and abstract axiomatization of the 'quantum information-flow' which enables protocols such as quantum teleportation. To this means we define strongly compact closed categories which abstractly capture the behavioral properties of quantum entanglement. . 'Postulates for an abstract quantum formalism' in which classical information-flow (e.g. token exchange) is part of the formalism. As an example, we provide a purely formal description of quantum teleportation and prove correctness in abstract generality. In this formalism types reflect kinds contra the essentially typeless von Neumann formalism. Hence even concretely this formalism manifestly improves on the usual one. .'Towards a high-level approach to quantum informatics'. Indeed the above discussed work can be conceived as aiming to solve: ???/von Neumann quantum formalism ≈ high-level language/low-level language
The following notes introduce Quantum Mechanics at an advanced level addressing students of Physics, Mathematics, Chemistry and Electrical Engineering. The aim is to put mathematical concepts and techniques like the path integral, algebraic techniques, Lie algebras and representation theory at the readers disposal. For this purpose we attempt to motivate the various physical and mathematical concepts as well as provide detailed derivations and complete sample calculations. We have made every effort to include in the derivations all assumptions and all mathematical steps implied, avoiding omission of supposedly ‘trivial’ information. Much of the author’s writing effort went into a web of cross references accompanying the mathematical derivations such that the intelligent and diligent reader should be able to follow the text with relative ease, in particular, also when mathematically difficult material is presented. In fact, the author’s driving force has been his desire to pave the reader’s way into territories unchartered previously in most introductory textbooks, since few practitioners feel obliged to ease access to their field. Also the author embraced enthusiastically the potential of the TEX typesetting language to enhance the presentation of equations as to make the logical pattern behind the mathematics as transparent as possible. Any suggestion to improve the text in the respects mentioned are most welcome. It is obvious, that even though these notes attempt to serve the reader as much as was possible for the author, the main effort to follow the text and to master