Estimating quantum state purity is a key task for assessing decoherence and state-preparation fidelity in quantum experiments, yet full tomography becomes infeasible for large systems. We consider the problem of estimating the purities of multiple unknown quantum states using Pauli measurements, where each Pauli basis can be employed for up to $M$ measurements in total. In this setting, the central challenge is to allocate the limited measurement resources efficiently to enhance overall estimation precision. We propose an adaptive allocation strategy that dynamically redistributes measurements according to observed outcomes. Theoretically, our method achieves a worst-case error scaling of $O\left(M^{-3/2}\right)$, outperforming nonadaptive approaches limited to $O\left(M^{-1}\right)$. Numerical simulations confirm the predicted scaling and robustness across different state ensembles. These results establish an efficient and scalable framework for multistate purity estimation, with direct relevance to quantum property learning and noise characterization in noisy intermediate-scale quantum devices.

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• Quantum information theory