Purity of Density Operators for Non-Orthogonal Quantum States with Bounds and Applications to Quantum Fisher Information

In this study, we investigate the purity of density operators constructed from non-orthogonal quantum states. Starting from a general statistical mixture of normalized states, we derive an explicit expression for the purity in terms of the associated probability distribution and the pairwise overlaps between the states, quantified by |⟨ψi|ψj⟩|2. This formulation clearly separates the contributions arising from classical probability weights and quantum interference effects induced by non-orthogonality. To further analyze the impact of state overlaps, we introduce an average overlap quantity that captures the global contribution of non-orthogonal components. Based on this representation, we establish upper bounds for the purity using both global and local overlap parameters. In particular, we show that the purity can be effectively controlled by the maximum pairwise overlap, eliminating the need to compute all individual inner products explicitly. A refined bound is also obtained by incorporating local overlap measures, leading to a tighter characterization of the system. In addition, we develop a general framework connecting overlap-based purity bounds with quantum Fisher information (QFI) in parameter-dependent quantum systems. We show that the overlap structure can be used to indirectly control and estimate the behavior of QFI, especially in cases where direct spectral analysis is not tractable. The applicability of the proposed approach is demonstrated through an explicit example involving non-orthogonal qubit states. These results provide a practical framework for estimating the mixedness of quantum states and offer useful insights into the interplay between non-orthogonality, purity, and quantum Fisher information in quantum information theory.