Abordagem axiomática para medidas de correlações totais

Abstract

Resumo: Correlações são de extrema importância para a ciência. Quando tratamos do caso quântico, entretanto, não há consenso sobre a melhor forma de medir as correlações totais. Na literatura, a informação mútua quântica é comumente usada como a principal maneira de quantificar essas correlações, mas nesta dissertação apresentaremos novas abordagens. Introduziremos medidas baseadas em divergências, como as de Tsallis, Rényi e Kullback-Leibler, medidas baseadas em métricas de normas e, por fim, uma medida baseada em covariância. Para validar esses quantificadores, propomos cinco axiomas matemáticos que pontuamos como necessários que uma boa medida de correlação satisfaça. Nossos testes revelam que algumas das medidas propostas não satisfazem todos os axiomas enumerados. Enfatizamos também que, embora a informação mútua quântica, a norma da matriz de correlação e o coeficiente de correlação de Pearson exibam equivalências para sistemas de dois qubits, todas apresentam o tradicional problema de ordenamento, muito comum em medidas de emaranhamento, por exemplo

Abstract: Correlations are of utmost importance to science. However, when dealing with the quantum case, there is no consensus on the best way to measure total correlations. In the literature, quantum mutual information is commonly used as the primary method to quantify these correlations, but in this dissertation, we will present new approaches. We will introduce measures based on divergences, such as Tsallis, Rényi, and KullbackLeibler divergences, measures based on norm metrics, and finally, a measure based on covarianceCorrelations are of utmost importance to science. However, when dealing with the quantum case, there is no consensus on the best way to measure total correlations. In the literature, quantum mutual information is commonly used as the primary method to quantify these correlations, but in this dissertation, we will present new approaches. We will introduce measures based on divergences, such as Tsallis, Rényi, and KullbackLeibler divergences, measures based on norm metrics, and finally, a measure based on covariance. To validate these quantifiers, we introduce five mathematical axioms that we consider essential for a good correlation measure to satisfy. Our tests reveal that some of the proposed measures do not fulfill all the enumerated axioms. We also emphasize that, although quantum mutual information, the correlation matrix norm, and the Pearson correlation coefficient exhibit equivalences for two-qubit systems, all of them present the traditional ordering problem, which is commonly encountered in entanglement measures, for example. To validate these quantifiers, we introduce five mathematical axioms that we consider essential for a good correlation measure to satisfy. Our tests reveal that some of the proposed measures do not fulfill all the enumerated axioms. We also emphasize that, although quantum mutual information, the correlation matrix norm, and the Pearson correlation coefficient exhibit equivalences for two-qubit systems, all of them present the traditional ordering problem, which is commonly encountered in entanglement measures, for example