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Equivalence between divisibility and monotonic decrease of information in classical and quantum stochastic processes

Francesco Buscemi and Nilanjana Datta
Phys. Rev. A 93, 012101 – Published 4 January 2016
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  • INTRODUCTION
  • QUANTUM DYNAMICAL MAPPINGS
  • DIVISIBILITY AND THE MEMORYLESS PROPERTY
  • INFORMATION-DECREASING DYNAMICAL…
  • THE SEMICLASSICAL CASE
  • THE QUANTUM CASE
  • DISCUSSION
  • ACKNOWLEDGMENTS
    • References

    Abstract

    The crucial feature of a memoryless stochastic process is that any information about its state can only decrease as the system evolves. Here we show that such a decrease of information is equivalent to the underlying stochastic evolution being divisible. The main result, which holds independently of the model of the microscopic interaction and is valid for both classical and quantum stochastic processes, relies on a quantum version of the so-called Blackwell-Sherman-Stein theorem in classical statistics.

    • Figure
    • Figure
    • Received 20 November 2014
    • Revised 16 September 2015

    DOI:https://doi.org/10.1103/PhysRevA.93.012101

    ©2016 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Quantum communicationStochastic processes
    1. Techniques
    Markovian processes
    Atomic, Molecular & Optical

    Authors & Affiliations

    Francesco Buscemi1,* and Nilanjana Datta2

    • 1Graduate School of Information Science, Nagoya University, Chikusa-ku, Nagoya 464-8601, Japan
    • 2Statistical Laboratory, University of Cambridge, Cambridge CB3 0WB, United Kingdom

    • *Corresponding author: buscemi@is.nagoya-u.ac.jp

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    Issue

    Vol. 93, Iss. 1 — January 2016

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    • Figure 1
      Figure 1

      Schematic representation of a discrete-time dynamical mapping. Top: The system's evolution from an initial time t0 to the final time tN is represented by a sequence of quantum channels (Ni)0≤i≤N (the thin arrows), describing the stochastic evolution of the system from the initial time t0 to some later time ti≥t0. Bottom: The dynamical mapping is divisible if there exists another sequence of quantum channels (Ci)1≤i≤N (the thick arrows) such that Ni+1=Ci+1∘Ni for all 0≤i≤N−1.

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    • Figure 2
      Figure 2

      Top: The joint system-environment evolution. Middle: The reduced dynamics of the system is divisible if it cannot be distinguished from a sequence of independent quantum channels (Ci)i≥1 applied in series. Bottom: As a consequence of the Stinespring-Kraus representation theorem [29, 30], any divisible process can always be thought of as originating from a collisionlike model, in which the system interacts with an environment that is reset after every interaction.

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