New entropic inequalities for qubit and unimodal Gaussian states

• Published in: arXiv.org 2017-09
• Main Authors: López-Saldívar, J A, Castaños, O, Man'ko, M A, Man'ko, V I
• Format: Article
• Language: English
• Subjects: Dependence; Entropy; Mathematical analysis; Parametric amplifiers; Partitions; Partitions (mathematics)
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1709.07256

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