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

Let Mn() denote the set of n by n complex matrices. Consider continuous time quantum semigroups t = et, t ≥ 0, where : Mn() → Mn() is the infinitesimal generator. If we assume that (I) = 0, we will call et, t ≥ 0 a quantum Markov semigroup. Given a stationary density matrix ρ = ρ, for the quantum Markov semigroup t, t ≥ 0, we can define a continuous time stationary quantum Markov process, denoted by Xt, t ≥ 0. Given an a priori Laplacian operator 0: Mn() → Mn(), we will present a natural concept of entropy for a class of density matrices on Mn(). Given a Hermitian operator A: n → n (which plays the role of a Hamiltonian), we will study a version of the variational principle of pressure for A. A density matrix ρA maximizing pressure will be called an equilibrium density matrix. From ρA we will derive a new infinitesimal generator A. Finally, the continuous time quantum Markov process defined by the semigroup t = etA, t ≥ 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.