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

Abstract

Let [Formula: see text] denote the set of [Formula: see text] by [Formula: see text] complex matrices. Consider continuous time quantum semigroups [Formula: see text], [Formula: see text], where [Formula: see text] is the infinitesimal generator. If we assume that [Formula: see text], we will call [Formula: see text], [Formula: see text] a quantum Markov semigroup. Given a stationary density matrix [Formula: see text], for the quantum Markov semigroup [Formula: see text], [Formula: see text], we can define a continuous time stationary quantum Markov process, denoted by [Formula: see text], [Formula: see text] Given an a priori Laplacian operator [Formula: see text], we will present a natural concept of entropy for a class of density matrices on [Formula: see text]. Given a Hermitian operator [Formula: see text] (which plays the role of a Hamiltonian), we will study a version of the variational principle of pressure for [Formula: see text]. A density matrix [Formula: see text] maximizing pressure will be called an equilibrium density matrix. From [Formula: see text] we will derive a new infinitesimal generator [Formula: see text]. Finally, the continuous time quantum Markov process defined by the semigroup [Formula: see text], [Formula: see text], and an initial stationary density matrix, will be called the continuous time equilibrium quantum Markov process for the Hamiltonian [Formula: see text]. It corresponds to the quantum thermodynamical equilibrium for the action of the Hamiltonian [Formula: see text].