Thermodynamic formalism for continuous-time quantum Markov semigroups: the detailed balance condition, entropy, pressure and equilibrium quantum processes

$M_n(\mathbb{C})$ denotes the set of $n$ by $n$ complex matrices. Consider continuous time quantum semigroups $\mathcal{P}_t= e^{t\, \mathcal{L}}$, $t \geq 0$, where $\mathcal{L}:M_n(\mathbb{C}) \to M_n(\mathbb{C})$ is the infinitesimal generator. If we assume that $\mathcal{L}(I)=0$, we will call $e^{t\, \mathcal{L}}$, $t \geq 0$ a quantum Markov semigroup. Given a stationary density matrix $\rho= \rho_{\mathcal{L}}$, for the quantum Markov semigroup $\mathcal{P}_t$, $t \geq 0$, we can define a continuous time stationary quantum Markov process, denoted by $X_t$, $t \geq 0.$ Given an {\it a priori} Laplacian operator $\mathcal{L}_0:M_n(\mathbb{C}) \to M_n(\mathbb{C})$, we will present a natural concept of entropy for a class of density matrices on $M_n(\mathbb{C})$. Given an Hermitian operator $A:\mathbb{C}^n\to \mathbb{C}^n$ (which plays the role of an Hamiltonian), we will study a version of the variational principle of pressure for $A$. A density matrix $\rho_A$ maximizing pressure will be called an equilibrium density matrix. From $\rho_A$ we will derive a new infinitesimal generator $\mathcal{L}_A$. Finally, the continuous time quantum Markov process defined by the semigroup $\mathcal{P}_t= e^{t\, \mathcal{L}_A}$, $t \geq 0$, and an initial stationary density matrix, will be called the continuous time equilibrium quantum Markov process for the Hamiltonian $A$. It corresponds to the quantum thermodynamical equilibrium for the action of the Hamiltonian $A$.