Quantum Bernoulli experiments and quantum stochastic processes

Manfred Wolff ()
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Manfred Wolff: Eberhard-Karls-Univ. Tübingen, Mathematisches Institut

Chapter 13 in The Strength of Nonstandard Analysis, 2007, pp 189-205 from Springer

Abstract: Abstract Based on a W*-algcbraic approach to quantum probability theory we construct basic discrete internal quantum stochastic processes with independent increments. We obtain a one-parameter family of (classical) Bernoulli experiments as linear combinations of these basic processes. Then we use the nonstandard hull of the internal GNS-Hilbert space $$ \mathcal{H}_\tau $$ corresponding to the chosen state τ (the underlying quantum probability measure) in order to derive nonstandard hulls of our internal processes. Finally continuity requirements lead to the specification of a certain subspace $$ \mathcal{L} $$ of to which the nonstandard hulls of our internal processes can be restricted and which turns out to be isomorphic to the Loeb-Guichardet space introduced by Leitz-Martini [10]. A subspace of $$ \mathcal{L} $$ then is shown to be isomorphic to the symmetric Fock space $$ \mathcal{F}_ + $$ (L 2([0,1], λ)) and our basic processes agree with the processes of Hudson and Parthasarathy on this subspace.

Keywords: Quantum Probability; Strong Operator Topology; Symmetric Tensor Product; Quantum Random Walk; Nonstandard Hull (search for similar items in EconPapers)
Date: 2007
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-211-49905-4_13

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DOI: 10.1007/978-3-211-49905-4_13

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