In recent years, biodiversity measures have gained prominence as essential tools for ecological and environmental assessments, particularly in the context of increasingly complex and large-scale datasets. We provide a comprehensive review of diversity measures, including the Gini-Simpson index, Hill numbers, and Rao's quadratic entropy, examining their roles in capturing various aspects of biodiversity. Among these, Rao's quadratic entropy stands out for its ability to incorporate not only species abundance but also functional and genetic dissimilarities. The paper emphasizes the statistical and ecological significance of Rao's quadratic entropy under the information geometry framework. We explore the distribution maximizing such a diversity measure under linear constraints that reflect ecological realities, such as resource competition or habitat suitability. Furthermore, we discuss a unified approach of the Leinster-Cobbold index combining Hill numbers and Rao's entropy, allowing for an adaptable and similarity-sensitive measure of biodiversity. Finally, we discuss the information geometry associated with the maximum diversity distribution focusing on the cross diversity measures such as the cross-entropy.

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