Eike Neumann ; Martin Pape ; Thomas Streicher - Computability in Basic Quantum Mechanics

lmcs:3222 - Logical Methods in Computer Science, June 19, 2018, Volume 14, Issue 2 - https://doi.org/10.23638/LMCS-14(2:14)2018
Computability in Basic Quantum Mechanics

Authors: Eike Neumann ; Martin Pape ; Thomas Streicher

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.

Volume: Volume 14, Issue 2
Published on: June 19, 2018
Accepted on: June 19, 2018
Submitted on: March 28, 2017
Keywords: Computer Science - Logic in Computer Science,03B70, 03F60, 18C50, 68Q55


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