Quantum detailed balance conditions are often formulated as relationships between a 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 $\B$ satisfying a quantum detailed balance condition when the duality is defined by means of pre-scalar products on $\B$ of the form $\langle x,y\rangle_s:=\tr(\rho^{1-s}x^*\rho^sy)$ ($s\in[0,1]$) in order to compare the resulting quantum versions of the classical detailed balance condition. Moreover, we discuss the structure of generators of Quantum Markov Semigroup which commute with the modular automorphism because this condition appears when we consider pre-scalar products with $s\not=1/2$.

### On two quantum versions of the detailed balance condition

#### Abstract

Quantum detailed balance conditions are often formulated as relationships between a 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 $\B$ satisfying a quantum detailed balance condition when the duality is defined by means of pre-scalar products on $\B$ of the form $\langle x,y\rangle_s:=\tr(\rho^{1-s}x^*\rho^sy)$ ($s\in[0,1]$) in order to compare the resulting quantum versions of the classical detailed balance condition. Moreover, we discuss the structure of generators of Quantum Markov Semigroup which commute with the modular automorphism because this condition appears when we consider pre-scalar products with $s\not=1/2$.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11567/517119
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