Thermodynamic Formalism for Continuous-Time Quantum Markov Semigroups: the Detailed Balance Condition, Entropy, Pressure and Equilibrium Quantum Processes

Let $M_n(ℂ)$ denote the set of $n$ by $n$ complex matrices. Consider continuous time quantum semigroups ${{𝒫}}_t=e^{t{ℒ}}$, $t\ge 0$, where ${ℒ}:M_n(ℂ)\to M_n(ℂ)$ is the infinitesimal generator. If we assume that ${ℒ}(I)=0$, we will call $e^{t{ℒ}}$, $t\ge 0$ a quantum Markov semigroup. Given a stationary density matrix $\rho ={\rho }_{{ℒ}}$, for the quantum Markov semigroup ${{𝒫}}_t$, $t\ge 0$, we can define a continuous time stationary quantum Markov process, denoted by $X_t$, $t\ge 0.$ Given an a priori Laplacian operator ${{ℒ}}_0:M_n(ℂ)\to M_n(ℂ)$, we will present a natural concept of entropy for a class of density matrices on $M_n(ℂ)$. Given a Hermitian operator $A:{ℂ}^n\to {ℂ}^n$ (which plays the role of a 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 ${{ℒ}}_A$. Finally, the continuous time quantum Markov process defined by the semigroup ${{𝒫}}_t=e^{t{{ℒ}}_A}$, $t\ge 0$, 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 $A$.