New entropic inequalities for qubit and unimodal Gaussian states

TL;DR

This paper derives new entropic inequalities for qubit and Gaussian states using approximations of the Tsallis relative entropy, leading to insights on state distances, energy, and entropy relations in quantum systems.

Contribution

It introduces a novel inequality connecting energy, entropy, and partition function based on von Neumann entropy, and applies it to qubit and Gaussian states.

Findings

Derived an approximation for Tsallis relative entropy near q=1.

Established a new inequality relating energy, entropy, and partition function.

Analyzed the distance measure between states for qubits and 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 $\hat{\sigma}=e^{- \beta \hat{H}}/{\rm Tr}(e^{- \beta \hat{H}})$ 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.