Researchers have introduced new estimators for quantum Rényi and Tsallis entropies, achieving nearly optimal sample complexity. These estimators are crucial for quantifying entanglement and uncertainty in quantum systems. The sample complexity, which measures the number of measurements required to achieve a given precision, has been significantly improved compared to previous works, approaching fundamental theoretical limits. This advancement is particularly relevant in the context of quantum information, where precise characterization of quantum states is a computational challenge.

For entropy orders α between 0 and 1, the sample complexity for Rényi entropy is O(d^(1+1/α)/ε^(1/α) + d^(1/α-1)/ε^2), and for Tsallis entropy it is O(d^(1+1/α)/ε^(1/α) + d^(2-2α)/ε^2), where d is the system dimension and ε is the additive error. Specifically, for 0 < α ≤ 1/2, the complexity for both entropies simplifies to O(d^(1+1/α)/ε^(1/α)). For non-integer α > 1, the sample complexity for Rényi entropy is O(d^2/ε^(1/α) + d^(1-1/α)/ε^2).

These new upper bounds represent a substantial improvement over previous estimators, such as those proposed by Acharya, Issa, Shende, and Wagner (2017) for Rényi entropy, and by Chen, Liu, and Wang (2026) for Tsallis entropy. Furthermore, the obtained results match the lower bounds recently established by Wang (2026), confirming the nearly optimal efficiency of the estimators. This achievement has direct implications for the design of experiments in quantum computing and quantum metrology, where efficiency in data collection is a critical factor.