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    Markov chains with doubly stochastic transition matrices and application to a sequence of non-selective quantum measurements

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    Publication date
    2022-05
    End of Embargo
    2023-10-14
    Author
    Vourdas, Apostolos
    Keyword
    Markov chains with doubly stochastic transition matrices
    Non-selective quantum measurements
    Rights
    © 2022 Elsevier. Reproduced in accordance with the publisher's self-archiving policy. This manuscript version is made available under the CC-BY-NC-ND 4.0 license.
    Peer-Reviewed
    yes
    Open Access status
    Embargoed access
    
    Metadata
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    Abstract
    A time-dependent finite-state Markov chain that uses doubly stochastic transition matrices, is considered. Entropic quantities that describe the randomness of the probability vectors, and also the randomness of the discrete paths, are studied. Universal convex polytopes are introduced which contain all future probability vectors, and which are based on the Birkhoff–von Neumann expansion for doubly stochastic matrices. They are universal in the sense that they depend only on the present probability vector, and are independent of the doubly stochastic transition matrices that describe time evolution in the future. It is shown that as the discrete time increases these convex polytopes shrink, and the minimum entropy of the probability vectors in them increases. These ideas are applied to a sequence of non-selective measurements (with different projectors in each step) on a quantum system with -dimensional Hilbert space. The unitary time evolution in the intervals between the measurements, is taken into account. The non-selective measurements destroy stroboscopically the non-diagonal elements in the density matrix. This ‘hermaphrodite’ system is an interesting combination of a classical probabilistic system (immediately after the measurements) and a quantum system (in the intervals between the measurements). Various examples are discussed. In the ergodic example, the system follows asymptotically all discrete paths with the same probability. In the example of rapidly repeated non-selective measurements, we get the well known quantum Zeno effect with ‘frozen discrete paths’ (presented here as a biproduct of our general methodology based on Markov chains with doubly stochastic transition matrices).
    URI
    http://hdl.handle.net/10454/18802
    Version
    Accepted manuscript
    Citation
    Vourdas A (2022) Markov chains with doubly stochastic transition matrices and application to a sequence of non-selective quantum measurements. Physica A: Statistical Mechanics and its Applications. 593: 126911
    Link to publisher’s version
    https://doi.org/10.1016/j.physa.2022.126911
    Type
    Article
    Notes
    The full text will be available at the end of the publisher's embargo: 14th Oct 2023
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    Engineering and Informatics Publications

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