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Quantum Quadratic Stochastic Processes and Their Ergodic Properties

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Quantum Quadratic Operators and Processes

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2133))

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Abstract

Quadratic stochastic processes describe the physical systems defined above, but they do not cover the cases at a quantum level. So, it is natural to define a concept of a quantum quadratic process. In this chapter we will define a quantum (noncommutative) analogue of quadratic stochastic processes. In our case, such a process will be defined on a von Neumann algebra, and we will study its ergodic properties such as the ergodic principle, regularity, etc. From the physical point of view, such a principle means that for sufficiently large values of time a system described by the process does not depend on the initial state of the system.

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Notes

  1. 1.

    Note that here \(\ell^{\infty }(E) \otimes \mathcal{M}\) is identified with \(\ell^{\infty }(E; \mathcal{M})\).

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Authors and Affiliations

  1. Dept. of Comput. & Theor. Sciences, International Islamic University Malaysia, Kuantan, Malaysia

    Farrukh Mukhamedov & Nasir Ganikhodjaev

Authors
  1. Farrukh Mukhamedov
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  2. Nasir Ganikhodjaev
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Mukhamedov, F., Ganikhodjaev, N. (2015). Quantum Quadratic Stochastic Processes and Their Ergodic Properties. In: Quantum Quadratic Operators and Processes. Lecture Notes in Mathematics, vol 2133. Springer, Cham. https://doi.org/10.1007/978-3-319-22837-2_8

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