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

TL;DR

This paper introduces a measure called intrinsic density of quantum states (IDQS) in information geometry, exploring its implications for quantum measurement and multi-parameter estimation, and revealing connections with uncertainty and Berry curvature.

Contribution

It defines IDQS as a high-dimensional measure in quantum information geometry and applies it to analyze quantum measurement limits and multi-parameter estimation.

Findings

IDQS measures the density of quantum states in projective Hilbert spaces.

A determinant form of quantum Cramér-Rao inequality is proposed.

The gap between IDQS and maximal distinguishable states relates to Berry curvature.

Abstract

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