  The Tsallis relative entropy $S_q (\hat{\rho},\hat{\sigma})$ measures the
distance between two arbitrary density matrices $\hat{\rho}$ and
$\hat{\sigma}$. In this work the approximation to this quantity when
$q=1+\delta$ ($\delta\ll 1$) is obtained. It is shown that the resulting series
is equal to the von Neumann relative entropy when $\delta=0$. Analyzing the von
Neumann relative entropy for arbitrary $\hat{\rho}$ and a thermal equilibrium
state $\hat{\sigma}=e^{- \beta \hat{H}}/{\rm Tr}(e^{- \beta \hat{H}})$ is
possible to define a new inequality relating the energy, the entropy, and the
partition function of the system. From this inequality, a parameter that
measures the distance between the two states is defined. This distance is
calculated for a general qubit system and for an arbitrary unimodal Gaussian
state. In the qubit case, the dependence on the purity of the system is studied
for $T \geq 0$ and also for $T<0$. In the Gaussian case, the general partition
function given a unimodal quadratic Hamiltonian is calculated and the
comparison of the thermal light state as a thermal equilibrium state of the
parametric amplifier is presented.
