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On the relationship between a quantum Markov semigroup and its representation via linear stochastic Schrödinger equations

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Abstract

A quantum Markov semigroup can be represented via classical diffusion processes solving a stochastic Schrödinger equation. In this paper we first prove that a quantum Markov semigroup is irreducible if and only if classical diffusion processes are total in the Hilbert space of the system. Then we study the relationship between irreducibility of a quantum Markov semigroup and properties of these diffusions such as accessibility, the Lie algebra rank condition, and irreducibility. We prove that all these properties are, in general, stronger than irreducibility of the quantum Markov semigroup, nevertheless, they are equivalent for some important classes of semigroups.

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Authors and Affiliations

  1. Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133, Milano, Italy

    Franco Fagnola

  2. CI2MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Barrio Universitario, Avenida Esteban Iturra s/n, 4089100, Casilla 160-C Concepción, Chile

    Carlos Mora

Authors
  1. Franco Fagnola
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  2. Carlos Mora
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Correspondence to Franco Fagnola.

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Dedicated to Prof. Kalyan B. Sinha on occasion of his 70th birthday.

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Fagnola, F., Mora, C. On the relationship between a quantum Markov semigroup and its representation via linear stochastic Schrödinger equations. Indian J Pure Appl Math 46, 399–414 (2015). https://doi.org/10.1007/s13226-015-0142-7

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  • DOI: https://doi.org/10.1007/s13226-015-0142-7

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