Quantum Itô algebra and quantum martingale

Volume 78 / 2007

Viacheslav Belavkin, Un Cig Ji Banach Center Publications 78 (2007), 47-58 MSC: Primary 81S25; Secondary 46F25. DOI: 10.4064/bc78-0-3


In this paper, we study a representation of the quantum Itô algebra in Fock space and then by using a noncommutative Radon-Nikodym type theorem we study the density operators of output states as quantum martingales, where the output states are absolutely continuous with respect to an input (vacuum) state. Then by applying quantum martingale representation we prove that the density operators of regular, absolutely continuous output states belong to the commutant of the $\star$-algebra parameterizing the quantum Itô algebra.


  • Viacheslav BelavkinMathematics Department
    University of Nottingham
    University Park
    Nottingham NG7 2RD, UK
  • Un Cig JiDepartment of Mathematics
    Research Institute of Mathematical Finance
    Chungbuk National University
    Cheongju 361-763, Korea

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