On two quantum versions of the detailed balance condition
Volume 89 / 2010
Abstract
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 $s\not=1/2$.
