New entropic inequalities for qubit and unimodal Gaussian states

The Tsallis relative entropy S-q((rho) over cap, (sigma) over cap) measures the distance between two arbitrary density matrices (rho) over cap and (sigma) over cap. In this work the approximation to this quantity when q = 1+ delta (delta << 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 an arbitrary (rho) over cap and a thermal equilibrium state (sigma) over cap = e(-beta(H) over cap)/Tr(e(-beta(H) over cap)) 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. (C) 2017 Elsevier B.V. All rights reserved.