On two quantum versions of the detailed balance condition

Tom 89 / 2010

Franco Fagnola, Veronica Umanità Banach Center Publications 89 (2010), 105-119 MSC: 46L55, 47D05, 82B10, 82C10, 81S25. DOI: 10.4064/bc89-0-5


Quantum detailed balance conditions are often formulated as relationships between the generator of a quantum Markov semigroup and the generator of a dual semigroup with respect to a certain scalar product defined by an invariant state. In this paper we survey some results describing the structure of norm continuous quantum Markov semigroups on ${\cal B}({\mathsf h})$ satisfying a quantum detailed balance condition when the duality is defined by means of pre-scalar products on ${\cal B}({\mathsf h})$ of the form $\langle x,y\rangle_s:=\mathop{\rm tr}(\rho^{1-s}x^*\rho^sy)$ ($s\in[0,1]$) in order to compare the resulting quantum versions of the classical detailed balance condition. Moreover, we discuss the structure of generators of a quantum Markov semigroup which commute with the modular automorphism because this condition appears when we consider pre-scalar products with $s\not=1/2$.


  • Franco FagnolaDipartimento di Matematica
    Politecnico di Milano
    Piazza Leonardo da Vinci 32
    I-20133 Milano, Italy
  • Veronica UmanitàDipartimento di Matematica
    Università di Genova
    Via Dodecaneso 35
    I-16146 Genova, Italy

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