New entropic inequalities for qubit and unimodal Gaussian states

The Tsallis relative entropy Sq(ρˆ,σˆ) measures the distance between two arbitrary density matrices ρˆ and σˆ. In this work the approximation to this quantity when q=1+δ (δ≪1) is obtained. It is shown that the resulting series is equal to the von Neumann relative entropy when δ=0. Analyzing the von Neumann relative entropy for an arbitrary ρˆ and a thermal equilibrium state σˆ=e−βHˆ∕Tr(e−βHˆ) 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≥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.