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arXiv:1610.09209 · Logical Methods in Computer Science

Computability in Basic Quantum Mechanics

Eike Neumann, Martin Pape, Thomas Streicher

Published Jun 19, 2018• 1 citation
arXivHTMLPDF

Abstract

The basic notions of quantum mechanics are formulated in terms of separable infinite dimensional Hilbert space $\mathcal{H}$. In terms of the Hilbert lattice $\mathcal{L}$ of closed linear subspaces of $\mathcal{H}$ the notions of state and observable can be formulated as kinds of measures as in [21]. The aim of this paper is to show that there is a good notion of computability for these data structures in the sense of Weihrauch's Type Two Effectivity (TTE) [26]. Instead of explicitly exhibiting admissible representations for the data types under consideration we show that they do live within the category $\mathbf{QCB}_0$ which is equivalent to the category $\mathbf{AdmRep}$ of admissible representations and continuously realizable maps between them. For this purpose in case of observables we have to replace measures by valuations which allows us to prove an effective version of von Neumann's Spectral Theorem.

DOI
10.23638/lmcs-14(2:14)2018

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The basic notions of quantum mechanics are formulated in terms of separable infinite dimensional Hilbert space H\mathcal{H}H. In terms of the Hilbert lattice L\mathcal{L}L of closed linear subspaces of H\mathcal{H}H the notions of state and observable can be formulated as kinds of measures as in [21]. The aim of this paper is to show that there is a good notion of computability for these data structures in the sense of Weihrauch's Type Two Effectivity (TTE) [26]. Instead of explicitly exhibiting admissible representations for the data types under consideration we show that they do live within the category QCB0\mathbf{QCB}_0QCB0​ which is equivalent to the category AdmRep\mathbf{AdmRep}AdmRep of admissible representations and continuously realizable maps between them. For this purpose in case of observables we have to replace measures by valuations which allows us to prove an effective version of von Neumann's Spectral Theorem.
``Quantum'' stands for for the concepts (both operational and formal) which had to be added to classical physics in order to understand otherwise unexplainable observed phenomena such as the structure of the spectral lines in atomic spectra. While the basic part of classical mechanics deals with the (essentially) reversible dynamics, quantum required adding the notions of ``measurement'' and (possibly non-local) ``correlations'' to the discussion. Crucially, all this comes with a ``probabilistic calculus''. The corresponding mathematical formalism was considered to have reached maturity in [von Neumann 1932], but there are some manifest problems with that formalism: (i) While measurements are applied to physical systems, application of their formal counterpart (i.e. a self-adjoint linear operator) to the vector representing that state of the system in no way reflects how the state changes during the act of measurement. Analogously, the composite of two self-adjoint operators has no physical significance while in practice measurements can be effectuated sequentially. More generally, the formal types in von Neumann's formalism do not reflect the nature of the corresponding underlying concept at all! (ii) Part of the problem regarding the measurements discussed above is that in the von Neumann formalism there is no place for storage, manipulation and exchange of the classical data obtained from measurements. Protocols such as quantum teleportation involving these cannot be given a full formal description. (iii) The behavioral properties of quantum entanglement which for example enable continuous data exchange using only finitary communication are hidden in the formalism. In [Abramsky and Coecke 2004] we addressed all these problems, and in addition provided a purely categorical axiomatization of quantum mechanics. The concepts of the abstract quantum mechanics are formulated relative to a strongly compact closed category with biproducts (of which the category FdHilb of finite dimensional Hilbert spaces and linear maps is an example). Preparations, measurements, either destructive or not, classical data exchange are all morphisms in that category, and their types fully reflect their kinds. Correctness properties of standard quantum protocols can be abstractly proven. Surprisingly, in this seemingly purely qualitative setting even the quantitative Born rule arises, that is the rule which tells you how to calculate the probabilities. Indeed, each such category has as endomorphism Hom of the tensor unit an abelian semiring of `scalars', and a special subset of these scalars will play the role of weights: each scalar induces a natural transformation which propagates through physical processes, and when a `state' undergoes a `measurement', the composition of the corresponding morphisms gives rise to the weight. Hence the probabilistic weights live within the category of processes. J. von Neumann. Mathematische Grundlagen der Quantenmechanik. Springer-Verlag (1932). English translation in Mathematical Foundations of Quantum Mechanics. Princeton University Press (1955). S. Abramsky and B. Coecke. A categorical semantics of quantum protocols. In the proceedings of LiCS'04 (2004). An extended version is available at arXiv:quant-ph/0402130 A more reader friendly version entitled `Quantum information flow, concretely, abstractly' is at http://www.vub.ac.be/CLEA/Bob/Papers/QPL.pdf
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