New entropic inequalities for qubit and unimodal Gaussian states

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.