A cycle decomposition and entropy production for circulant quantum Markov semigroups (Q2842036)

For the uniformly continuous quantum Markov semigroups (QMS) on the predual of \(\mathcal B(H)\), the authors put forward a modified notion of quantum entropy production rate (QEPR) with respect to a stationary state \(\rho\) (cf. [\textit{F. Fagnola} and \textit{R. Rebolledo}, in: Quantum probability and infinite dimensional analysis. Proceedings of the 29th conference, Hammamet, Tunisia, October 13--18, 2008. Ouerdiane, Habib (ed.) et al., Hackensack, NJ: World Scientific. QP--PQ: Quantum Probability and White Noise Analysis 25, 245--261 (2010; Zbl 1220.81151)]). In the case of \(\dim\mathcal B(H)<\infty\), they introduce a class of circulant QMS and discuss the relations to the embedded classical Markov chains generated by circulant matrices as well as a construction of special cycle representations (for the latter, see also [\textit{F. Fagnola} and \textit{V. Umanità}, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 15, No. 3, Paper No. 1250016, 17 p. (2012; Zbl 1257.82067)]). For that class, the QEPR is calculated explicitly and its vanishing turns out to be equivalent to a certain quantum detailed balance condition.