Bob Coecke

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Main topic:

• [The Logic of Entanglement. An invitation] - research report (2003) 160 pages. [The Logic of Entanglement] - 8 page short version of the above one. [Quantum information-flow, concretely, and axiomatically] - lecture notes of talks given between Summer 2003 and Spring 2004; they appeared as conference proceedings of both QPLII (2004) and QI-2 (2004); they also provide a physicist-friendly passage to the categorical quantum axiomatics
• [A categorical semantics of quantum protocols] - paper with Abramsky (IEEE-LiCS 2004) in which we initiated the categorical quantum mechanical formalism in terms of strongly compact closed categories. It provides an abstract counterpart to the logic of entanglement.
• [De-linearizing Linearity I: Projective Quantum Axiomatics from Strong Compact Closure] (QPL III, 2005). Here we establish a passage from vector-space-like categories to projective-space-like categories, by relaxing the biproduct structure.
• [Kindergarten Quantum Mechanics] lecture notes (2005) of talks at Google, PI workshop, QTRF-III and Kestrel Institute. (on certain screens the .ps and derived .pdf version at the arXiv shows light grey frames within certain pictures, if so, download the original .pdf source [here] in which also some typos are corrected)
• [Introducing Categories to the Practicing Physicist] - lecture notes (2005). These provide a very informal introduction to monoidal categories, but at the same time I make the point that monoidal categories are the most natural mathematical model for any physical theory. I still need to add some extra references and pointers to related stuff, correct some typos etc.
• [Quantum measurements without sums] - paper (2006) with [Dusko Pavlovic] (Kestrel Institute). We provide a purely multiplicative, ie purely tensor based, ie purely graphical account on (projective) quantum measurements, and on classicality and classical-quantum interaction. One of the nice features is the fact that von Neumann's projection postulate, in well-typed and resource-sensitive form, becomes the Eilenbergh-Moore coalgebraic condition for the functor X(x)- where X is a so-called classical object. [POVMs and Naimark's theorem without sums] - paper (2006) with [Eric Oliver Paquette] (Universite de Montreal). We give a purely graphical/multiplicative axiomatization of POVMs which is supported by a purely graphical counterpart to Naimark's theorem.

• [Physical Traces: Quantum vs. Classical Information Processing] - paper with Abramsky (CTCS 2002 & ENTCS 69, 2003) containing the first investigations in connections between quantum entanglement and certain monoidal structures; now outdated by [Abstract Physical Traces] - paper with Abramsky (TAC 14, 2005). See also [Strongly Compact Closed Semantics] (MFPS XXI & ENTCS 155, 2005) for more pictures, and the position statement with Abramsky (2005) entitled [Physics from Computer Science], an invited contribution to International Journal for Unconventional Computing.
• Slides of talks given between summer 2004 and winter 2005 mostly to non-physicists: [Abstract Quantum Mechanics] (complex phases are ignored in this one!). Slides of a talk at MFPS XXI in Spring 2005: [Strongly Compact Closed Quantum Semantics]; Slides of talk in July 2005: [De-linearizing Linearity] at QPL III; Slides [Category Theory Introduction] at a 2005 Perimeter Institute summer school. A [video recording] of my [Kindergarten Quantum Mechanics] talk at that event is available by typing kindergarten in the search window after selecting Quantum Information, Computation and Logic Workshop: Exploring New Connections. Here's another [Video recording] of a talk at The Impact of Categories: 60 Years of Category Theory in Historical and Philosophical Retrospect, Ecole Normale Superieure, Paris, October 10-14, 2005, with [From Coherence to Quantum] as title. The talks [In the Beginning God Created Tensor . . . as a Picture] and [Axiomatic Description of Mixed States] cover most of the things I've been talking about spring/summer 2006.

More related stuff is available from [Samson Abramsky]'s and [Peter Selinger]'s respective homepages, in [Houston]'s and [Svetlichny]'s papers and in particular [Vicary's recent one on the quantum harmonic oscillator] and [Ross Duncan]'s PhD-thesis entitled [Types for Quantum Computing]. Similar ideas can also be found in [John Baez]'s and [Louis Kauffman]'s papers. Also my students [Bill Edwards] and [Benjamin Jackson], and also [Colin Stephen] are working in this field. John Baez recently started group blog postings on the subject of [Quantum computation and symmetric monoidal categories]. This group blog is embedded within the [n-category cafe] which gives some nice reading on physics-meets-logic-meets-philosophy-meets-computation-meets-categories, which due to John's contribution is is particularly well-suited for physicists learing something about conceptually measningful structures. As for some history, graphical calculi in physics trace back to a 1971 paper by Penrose entitled Applications of Negative Dimensional Tensors - which is not on Quantum Mechanics but on General Relativity. Full and faithfull categorical semantics for these kinds of calculi was provided by Joyal and Street, and Freyd and Yetter, and implicitly also already in the pioneering Kelly and Laplaza paper entitled Coherence for Compact Closed Categories. Also closely related is the Carboni and Walters paper entitled Catesian Bicategories I. The connections to logic are mainly due to Lambek. One could argue that Deligne's paper entitled Catgories Tannakiennes of the 1990 Grothendieck Festschrift provides a representation theorem.