Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Measure of the density of quantum states in information geometry and its
  application in the quantum multi-parameter estimation   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.

20
The categories of this paper are: acc-phys
30
The categories of this paper are: q-alg
23
The categories of this paper are: solv-int
28
The categories of this paper are: alg-geom
3
The categories of this paper are: physics
34
The categories of this paper are: chem-ph
32
The categories of this paper are: atom-ph
31
The categories of this paper are: ao-sci
10
The categories of this paper are: q-bio
14
The categories of this paper are: math-ph
11
The categories of this paper are: hep-lat
26
The categories of this paper are: bayes-an
27
The categories of this paper are: comp-gas
19
The categories of this paper are: eess
36
The categories of this paper are: cmp-lg
18
The categories of this paper are: econ