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Quantum Communication, Computing, and Measurement pp 355–363Cite as

Time-Ordered Wick Exponential and Quantum Stochastic Differential Equations

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Abstract

Let H = L 2(ℝ,dt) be the real Hilbert space of L 2-functions on ℝ and let Γ(H ℂ) be the Boson Fock space over Hℂ, the complexification of H. Let ℌ be another complex Hilbert space. Given a one-parameter family of operators {L t} acting in Γ(H ℂ) ⊗ ℌ, we consider the initial value problem of the form:

$$ \frac{{d{\Xi _t}}}{{dt}}\; = \;{L_t}\;\diamondsuit \;{\Xi _t},\;\;\;\;{\left. \Xi \right|_{t = 0}}\; = \;{\Xi _0}$$
(1)

where ◊ is the Wick product. The main purpose of this paper is to prove the unique existence of a solution to equation (1) in the sense of distributions. The main statement will be found in §5.

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Author information

Authors and Affiliations

  1. Graduate School of Polymathematics, Nagoya University, Nagoya, 464-01, Japan

    Nobuaki Obata

Authors
  1. Nobuaki Obata
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Editor information

Editors and Affiliations

  1. Tamagawa University, Machida, Tokyo, Japan

    O. Hirota

  2. Steklov Mathematical Institute, Moscow, Russia

    A. S. Holevo

  3. University of New Mexico, Albuquerque, New Mexico, USA

    C. M. Caves

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© 1997 Springer Science+Business Media New York

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Obata, N. (1997). Time-Ordered Wick Exponential and Quantum Stochastic Differential Equations. In: Hirota, O., Holevo, A.S., Caves, C.M. (eds) Quantum Communication, Computing, and Measurement. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-5923-8_37

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  • DOI: https://doi.org/10.1007/978-1-4615-5923-8_37

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-7716-0

  • Online ISBN: 978-1-4615-5923-8

  • eBook Packages: Springer Book Archive

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