  In this work we present a formalism to describe non equilibrium conditions in
systems with a discretized energy spectrum, such as quantum systems. We develop
a formalism based on a combination of Gibbs-Shannon entropy and information
thermodynamics that arrives to a generalization of the De-Brujin identity
applicable to discrete and non-symmetric distributions. This allows to define
the concept of a thermodynamic temperature with a different, albeit
complementary meaning to the equilibrium kinetic temperature of a system. The
theory is applied to Bosonic and Fermionic cases represented by an harmonic
oscillator and a single energy state, respectively. We show that the formalism
correctly recovers known results at equilibrium, then we demonstrate an
application to a genuine non equilibrium state: a coherent quantum oscillator.
