New entropic inequalities for qubit and unimodal Gaussian states

Abstract

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 $\hat{\sigma }=e^{-\beta \hat{H}}/\mathrm{Tr}(e^{-\beta \hat{H}})$ 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 $T\ge 0$ 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.

Work accepted in Physica A: Statistical Mechanics and its Applications, In press

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