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Recently, there is a growing interest in study quantum mechanics from theinformation geometry perspective, where a quantum state is depicted with apoint in the projective Hilbert space. By taking quantum Fisher information(QFI) as the metric of projective Hilbert spaces, estimating a small parametershift is equivalent to distinguishing neighboring quantum states along a givencurve. Henceforth, information geometry plays a significant role in the singleparameter estimation. However, the absence of high dimensional measures limitsits applications in studying the multi-parameter systems. In this paper, wewill discuss the physical implications of the volume element of QFI. Itmeasures the intrinsic density of quantum states (IDQS) in projective Hilbertspaces, which is, then, a measure to define the (over) completeness relation ofa class of quantum states. As an application, IDQS can be used in quantummeasurement and multi-parameter estimation. We find the density ofdistinguishable states (DDS) for a set of efficient estimators is measured bythe invariant volume of the classical Fisher information, which is theclassical counterpart of QFI and serves as the metric of statistical manifolds.Correspondingly, a determinant form of quantum Cram\'{e}r-Rao inequality isproposed to quantify the ability to infer the IDQS via quantum measurement. Asa result, we find a gap between IDQS and maximal DDS over the measurements. Thegap has tight connections with the uncertainty relationship. Exemplified withthe three-level system with two parameters, we find the maximal DDS attainedvia the \emph{vertex measurements} (MvDDS) equals the square root of thequantum geometric tensor's determinant. It indicates the square gap betweenIDQS and MvDDS is proportional to the square of Berry curvature.

Measure of the density of quantum states in information geometry and its application in the quantum multi-parameter estimation