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Finite dimensional Hilbert spaces are complete for dagger compact closed categories
 In the proceedings of QPL 5
, 2008
"... We show that an equation follows from the axioms of dagger compact closed categories if and only if it holds in finite dimensional Hilbert spaces. Keywords: Dagger compact closed categories, Hilbert spaces, completeness. ..."
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We show that an equation follows from the axioms of dagger compact closed categories if and only if it holds in finite dimensional Hilbert spaces. Keywords: Dagger compact closed categories, Hilbert spaces, completeness.
A NonStandard Semantics for Kahn Networks in Continuous Time
"... In a seminal article, Kahn has introduced the notion of process network and given a semantics for those using Scott domains whose elements are (possibly infinite) sequences of values. This model has since then become a standard tool for studying distributed asynchronous computations. From the beginn ..."
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In a seminal article, Kahn has introduced the notion of process network and given a semantics for those using Scott domains whose elements are (possibly infinite) sequences of values. This model has since then become a standard tool for studying distributed asynchronous computations. From the beginning, process networks have been drawn as particular graphs, but this syntax is never formalized. We take the opportunity to clarify it by giving a precise definition of these graphs,
Fractional Types
"... Abstract. In previous work, we developed a firstorder, informationpreserving, and reversible programming language Π founded on type isomorphisms. Being restricted to firstorder types limits the expressiveness of the language: it is not possible, for example, to abstract common program fragments in ..."
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Abstract. In previous work, we developed a firstorder, informationpreserving, and reversible programming language Π founded on type isomorphisms. Being restricted to firstorder types limits the expressiveness of the language: it is not possible, for example, to abstract common program fragments into a higherlevel combinator. In this paper, we introduce a higherorder extension of Π based on the novel concept of fractional types 1/b. Intuitively, a value of a fractional type 1/v represents negative information. A function is modeled by a pair (1/v1, v2) with 1/v1 representing the needed argument and v2 representing the result. Fractional values are firstclass: they can be freely propagated and transformed but must ultimately — in a complete program — be offset by the corresponding amount of positive information. 1
unknown title
, 2009
"... A note on the biadjunction between 2categories of traced monoidal categories and tortile monoidal categories ..."
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A note on the biadjunction between 2categories of traced monoidal categories and tortile monoidal categories
c ○ 2009 Cambridge Philosophical Society
, 2009
"... categories and tortile monoidal categories ..."
QPL/DCM 2008
"... category is also dagger traced monoidal; conversely, by Joyal, Street, and Verity’s “Int ” construction, every dagger traced monoidal category can be fully embedded in a dagger compact closed category. We will make use of the soundness and completeness of the graphical representation, specifically o ..."
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category is also dagger traced monoidal; conversely, by Joyal, Street, and Verity’s “Int ” construction, every dagger traced monoidal category can be fully embedded in a dagger compact closed category. We will make use of the soundness and completeness of the graphical representation, specifically of the following result: Finite dimensional Hilbert spaces are complete for dagger compact closed categories (extended abstract)
Quantum picturalism
, 2009
"... Why did it take us 50 years since the birth of the quantum mechanical formalism to discover that unknown quantum states cannot be cloned? Yet, the proof of the ‘nocloning theorem’ is easy, and its consequences and potential for applications are immense. Similarly, why did it take us 60 years to dis ..."
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Why did it take us 50 years since the birth of the quantum mechanical formalism to discover that unknown quantum states cannot be cloned? Yet, the proof of the ‘nocloning theorem’ is easy, and its consequences and potential for applications are immense. Similarly, why did it take us 60 years to discover the conceptually intriguing and easily derivable physical phenomenon of ‘quantum teleportation’? We claim that the quantum mechanical formalism doesn’t support our intuition, nor does it elucidate the key concepts that govern the behaviour of the entities that are subject to the laws of quantum physics. The arrays of complex numbers are kin to the arrays of 0s and 1s of the early days of computer programming practice. Using a technical term from computer science, the quantum mechanical formalism is ‘lowlevel’. In this review we present steps towards a diagrammatic ‘highlevel ’ alternative for the Hilbert space formalism, one which appeals to our intuition. The diagrammatic language as it currently stands allows for intuitive reasoning about interacting quantum systems, and trivialises many otherwise involved and tedious computations. It clearly exposes limitations such as the nocloning theorem, and phenomena such as quantum teleportation. As a logic, it supports ‘automation’: it enables a (classical) computer to reason about interacting quantum systems, prove theorems, and design protocols. It allows for a wider variety of underlying theories, and can be easily modified, having the potential to provide the required stepstone towards a deeper conceptual understanding of quantum theory, as well as its