Quantum detailed balance conditions with time reversal: the finite-dimensional case

We classify generators of quantum Markov semigroups $\T$ on $\B$, with ${\mathsf{h}}$ finite-dimensional and with a faithful normal invariant state $\rho$, satisfying the standard quantum detailed balance condition with an anti-unitary time reversal $\theta$ commuting with $\rho$, namely $\tr(\rho^{1/2}x\rho^{1/2}\T_t(y)) = \tr(\rho^{1/2}\theta y^*\theta\rho^{1/2}\T_t(\theta x^*\theta))$ for all $x,y\in\B $ and $t\ge 0$. Our results also show that it is possible to find a standard form for the operators in the Lindblad representation of the generators extending the standard form of generators of quantum Markov semigroups satisfying the usual quantum detailed balance condition with non-symmetric multiplications $x\to \rho^{s}x\rho^{1-s}$ ($s\in [0,1]$, $s\not=1/2$) whose generators must commute with the modular group associated with $\rho$. This supports our conclusion that the most appropriate non-commutative version of the classical detailed balance condition is the above standard quantum detailed balance condition with an anti-unitary time reversal.