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Complementarity in categorical quantum mechanics
, 2010
"... We relate notions of complementarity in three layers of quantum mechanics: (i) von Neumann algebras, (ii) Hilbert spaces, and (iii) orthomodular lattices. Taking a more general categorical perspective of which the above are instances, we consider dagger monoidal kernel categories for (ii), so that ( ..."
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Cited by 23 (7 self)
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We relate notions of complementarity in three layers of quantum mechanics: (i) von Neumann algebras, (ii) Hilbert spaces, and (iii) orthomodular lattices. Taking a more general categorical perspective of which the above are instances, we consider dagger monoidal kernel categories for (ii), so that (i) become (sub)endohomsets and (iii) become subobject lattices. By developing a ‘pointfree’ definition of copyability we link (i) commutative von Neumann subalgebras, (ii) classical structures, and (iii) Boolean subalgebras.
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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Cited by 20 (4 self)
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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
Extending graphical representations for compact closed categories with applications to symbolic quantum computation
, 2008
"... Graphbased formalisms of quantum computation provide an abstract and symbolic way to represent and simulate computations. However, manual manipulation of such graphs is slow and error prone. We present a formalism, based on compact closed categories, that supports mechanised reasoning about such g ..."
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Cited by 5 (1 self)
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Graphbased formalisms of quantum computation provide an abstract and symbolic way to represent and simulate computations. However, manual manipulation of such graphs is slow and error prone. We present a formalism, based on compact closed categories, that supports mechanised reasoning about such graphs. This gives a compositional account of graph rewriting that preserves the underlying categorical semantics. Using this representation, we describe a generic system with a fixed logical kernel that supports reasoning about models of compact closed category. A salient feature of the system is that it provides a formal and declarative account of derived results that can include ‘ellipses’style notation. We illustrate the framework by instantiating it for a graphical language of quantum computation and show how this can be used to perform symbolic computation. Key words: graph rewriting, quantum computing, categorical logic, interactive theorem proving, graphical calculi.
Pictures of complete positivity in arbitrary dimension
 In Quantum Programming Languages, Electronic Proceedings in Theoretical Computer Science
, 2011
"... Two fundamental contributions to categorical quantum mechanics are presented. First, we generalize the CPconstruction, that turns any dagger compact category into one with completely positive maps, to arbitrary dimension. Second, we axiomatize when a given category is the result of this constructio ..."
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Cited by 5 (3 self)
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Two fundamental contributions to categorical quantum mechanics are presented. First, we generalize the CPconstruction, that turns any dagger compact category into one with completely positive maps, to arbitrary dimension. Second, we axiomatize when a given category is the result of this construction. 1
A graphical approach to measurementbased quantum computing
 Quantum Physics and Linguistics: A Compositional, Diagrammatic Discourse, chapter 3
, 2013
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QPL 2005 Preliminary Version Delinearizing Linearity: Projective Quantum Axiomatics from Strong Compact Closure
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"... This paper reports a use of flexible bodies in Multi Body System (MBS) modeling. A combination of discrete and modal flexibility enables structural eigenfrequency investigation. The Voluntary Milking System ™ (VMS) is a product from Swedish agricultural company DeLaval AB [1]. The VMS introduces an ..."
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This paper reports a use of flexible bodies in Multi Body System (MBS) modeling. A combination of discrete and modal flexibility enables structural eigenfrequency investigation. The Voluntary Milking System ™ (VMS) is a product from Swedish agricultural company DeLaval AB [1]. The VMS introduces an entirely different way of milking cows. The cows themselves decide when its time to be milked and make their way to the milking unit (the VMS). The VMS model will consist of three major part models: the robot arm, the stall, and the control system. These systems can be seen as sub models when using a systematic approach [5]. The modeling this far has been concentrated on the first two sub models, i.e. the robot arm and the stall. Covered here will be the finite element (FE) modeling activity, to represent structural elasticity, and the connection of the stall part model to the robot arm part model. Emphasis is on modeling activities. Preliminary results from simulations regarding forced oscillation are presented.