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• W62528541 abstract "``Quantum'' stands for for the concepts (both operational and formal)which had to be added to classical physics in order to understandotherwise unexplainable observed phenomena such as the structure ofthe spectral lines in atomic spectra. While the basic part ofclassical mechanics deals with the (essentially) reversibledynamics, quantum required adding the notions of ``measurement'' and(possibly non-local) ``correlations'' to the discussion. Crucially,all this comes with a ``probabilistic calculus''. The correspondingmathematical formalism was considered to have reached maturity in[von Neumann 1932], but there are some manifest problems with thatformalism:(i) While measurements are applied to physical systems, applicationof their formal counterpart (i.e. a self-adjoint linear operator) tothe vector representing that state of the system in no way reflectshow the state changes during the act of measurement. Analogously,the composite of two self-adjoint operators has no physicalsignificance while in practice measurements can be effectuatedsequentially. More generally, the formal types in von Neumann'sformalism do not reflect the nature of the corresponding underlyingconcept at all!(ii) Part of the problem regarding the measurements discussed aboveis that in the von Neumann formalism there is no place for storage,manipulation and exchange of the classical data obtained frommeasurements. Protocols such as quantum teleportation involvingthese cannot be given a full formal description.(iii) The behavioral properties of quantum entanglement which forexample enable continuous data exchange using only finitarycommunication are hidden in the formalism.In [Abramsky and Coecke 2004] we addressed all these problems, and inaddition provided a purely categorical axiomatization of quantummechanics. The concepts of the abstract quantum mechanics areformulated relative to a strongly compact closed category withbiproducts (of which the category FdHilb of finite dimensionalHilbert spaces and linear maps is an example). Preparations,measurements, either destructive or not, classical data exchange areall morphisms in that category, and their types fully reflect theirkinds. Correctness properties of standard quantum protocols can beabstractly proven.Surprisingly, in this seemingly purely qualitative setting even thequantitative Born rule arises, that is the rule which tells you howto calculate the probabilities. Indeed, each such category has asendomorphism Hom of the tensor unit an abelian semiring of`scalars', and a special subset of these scalars will play the roleof weights: each scalar induces a natural transformation whichpropagates through physical processes, and when a `state' undergoesa `measurement', the composition of the corresponding morphismsgives rise to the weight. Hence the probabilistic weights livewithin the category of processes.J. von Neumann. Mathematische Grundlagen der Quantenmechanik.Springer-Verlag (1932). English translation in MathematicalFoundations of Quantum Mechanics. Princeton University Press (1955).S. Abramsky and B. Coecke. A categorical semantics of quantumprotocols. In the proceedings of LiCS'04 (2004). An extended versionis available at arXiv:quant-ph/0402130 A more reader friendlyversion entitled `Quantum information flow, concretely, abstractly'is at http://www.vub.ac.be/CLEA/Bob/Papers/QPL.pdf" @default.
• W62528541 created "2016-06-24" @default.
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• W62528541 date "2005-01-01" @default.
• W62528541 modified "2023-09-26" @default.
• W62528541 title "Discrete classical vs. continuous quantum data in abstract quantum mechanics" @default.
• W62528541 hasPublicationYear "2005" @default.
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