Geometric relations in Classical and Quantum Information Theory using the Lambert-Tsallis Wq(x) function

Abstract

This work presents relations using the Lambert-Tsallis function Wq(x) that provide geometric characteristics in classical and quantum information theory. Firstly, the Wq(x) function is used to estimate parameters in discrete and continuous distributions knowing the a priori probability value. A lower bound for the probability density derived from the branch point of Wq(x). Secondly, we will represent the direct relationship between the Fisher distance in the HF2 hyperbolic model of normal distributions as a function of the distance associated with the Kulback-Leibler divergence in the argument of the Lambert-Tsallis function. Subsequently, using the disentropy based on Rényi, a functional that uses Wq(x) as its kernel, can be used as a measure of purity in the qubit state space (Bloch sphere), as well as for calculating quantum disentanglement of pure states of two qubits. Finally, representations for quantum fidelity and quantum affinity are defined as a function of Wq(x) once the Fisher and Wigner-Yanase quantum information metrics were known, connecting the Lambert-Tsallis function to the quantum speed limit (QSL) theory.