Coherent states in Bernoulli noise functionals. (English) Zbl 1220.60041

Summary: Let \((\Omega,{\mathcal F},\mathbb P)\) be a probability space and \(Z=(Z_K)_{k\in \mathbb N}\) a Bernoulli noise on \((\Omega,{\mathcal F},\mathbb P)\) which has the chaotic representation property. In this paper, we investigate a special family of functionals of \(Z\), which we call the coherent states. First, with the help of \(Z\), we construct a mapping \(\varphi \) from \(l^2(\mathbb N)\) to \(\mathbb L^2(\Omega,{\mathcal F},\mathbb P)\) which is called the coherent mapping. We prove that \(\varphi\) has the continuity property and other properties of operation. We then define functionals of the form \(\varphi (f)\) with \(f\in l^2(\mathbb N)\) as the coherent states and prove that all the coherent states are total in \({\mathcal L}^2(\Omega,{\mathcal F},\mathbb P)\). We also show that \(\varphi\) can be used to factorize \({\mathcal L}^2(\Omega,{\mathcal F},\mathbb P)\). Finally, we give an application of the coherent states to the calculus of quantum Bernoulli noise.


60H40 White noise theory
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
81S25 Quantum stochastic calculus
Full Text: DOI


[1] DOI: 10.1214/08-PS139 · Zbl 1189.60089
[2] Parthasarathy, An Introduction to Quantum Stochastic Calculus (1992) · Zbl 0751.60046
[3] Émery, Séminaire de Probabilités, XXXV pp 123– (2001)
[4] DOI: 10.1016/j.jmaa.2010.08.021 · Zbl 1205.60106
[5] DOI: 10.1063/1.3431028 · Zbl 1310.81107
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