Quantum Fisher information (QFI) is a key concept in quantum state estimation, and Berry curvature (BC) is another basic concept to describe geometric properties of quantum states. In this paper, we consider pure states undergoing unitary parametrization processes and show that the BC serves as a lower bound for the product of QFIs corresponding to two different parameters through the Heisenberg uncertainty relation. This relation between QFI and BC implies that the estimation precisions of two different parameters are mutually restrictive due to finite BC, and the notion of QFI squeezing is introduced. A scenario of general su(2) parametrization is considered in detail to verify the relation between the QFI and BC.
Quantum Fisher information (QFI) is a key concept in quantum state estimation, and Berry curvature (BC) is another basic concept to describe geometric properties of quantum states. In this paper, we consider pure states undergoing unitary parametrization processes and show that the BC serves as a lower bound for the product of QFIs corresponding to two different parameters through the Heisenberg uncertainty relation. This relation between QFI and BC implies that the estimation precisions of two different parameters are mutually restrictive due to finite BC, and the notion of QFI squeezing is introduced. A scenario of general su(2) parametrization is considered in detail to verify the relation between the QFI and BC.
Recently, there is growing interest in studying quantum mechanics from the information geometry perspective, where a quantum state is depicted by a point in the projective Hilbert space (PHS). However, the absence of high-dimensional measures limits information geometry in the study of multiparameter systems. In this paper, we propose a measure of the intrinsic density of quantum states (IDQS) in the PHS with the volume element of quantum Fisher information (QFI). Theoretically, the IDQS is a measure to define the (over)completeness relation of a class of quantum states. As an application, the IDQS is used to study quantum measurement and multiparameter estimation. We find that the density of distinguishable states (DDS) for a set of efficient estimators is measured by the invariant volume element of the classical Fisher information, which is the classical counterpart of the QFI and serves as the metric of statistical manifolds. The ability to infer the IDQS via quantum measurement is studied with a determinant-form quantum Cram\'er-Rao inequality. As a result, we find a gap between the IDQS and the maximal DDS over the measurements. The gap has tight connections with the uncertainty relationship. Exemplified by the three-level system with two parameters, we find that the Berry curvature characterizes the square gap between the IDQS and the maximal attainable DDS. Specific to vertex measurements, the square gap is proportional to the square of the Berry curvature.
For pure states, the quantum Berry curvature was well studied. However, the quantum curvature for mixed states has received less attention. From the concept of symmetric logarithmic derivative, we introduce a mixed-state quantum curvature and find that it plays a key role in the field of multi-parameter precision estimations. Through spectral decomposition, we derive the mixed-state Berry curvature for both the full-rank and non-full-rank density matrices. As an example, we obtain the exact expression of the Berry curvature for an arbitrary qubit state.
Abstract Quantum Fisher information (QFI) is a key concept of quantum metrology, which is widely used in quantum parameter estimation. We introduce a general approach to analytically get the QFI of pure states for unitary parametrization processes governed by arbitrary finite dimensional Lie algebras. We show the universality of this approach by comparing the result of algebra with the previous work and applying it to and algebras. For case, there are some interesting findings. The phase of the initial coherent state has an optimal value for maximum QFI, and the QFI can vanish and revive during the time evolution.
The efficiency of quantum measurements in distinguishing neighboring quantum states is a central topic in quantum information sciences. We study the efficiency of vertex measurement in two-parameter estimation from the information geometry perspective, where the vertex measurement is a rare typical measurement recognized as optimal for both single-parameter estimation and multiparameter estimation of states in manifolds with vanishing Berry curvature. This article focuses explicitly on scenarios with nonzero Berry curvature. We quantify the precision of an estimation scheme with the density of distinguishable states and measure the efficiency of a given vertex measurement by comparing its precision to the maximum value. Exemplified by the three-level system with two parameters, we show that the efficiency is highly anisotropic concerning the direction of a small mismatch vector between the applied measurement and the estimated parameter. It may even vanish in some specific directions. Furthermore, the direction is hardly acknowledged prior, hindering the practical application of vertex measurement in multiparameter estimations. To remove this obstacle, we further introduce a scheme that can adaptively approach the optimal vertex measurements.
Entanglement is widely regarded as an essential resource for a number of tasks and can in some cases be quantified by figures of merit related to those tasks. In quantum metrology, this is showcased by the connections between the quantum Fisher information (QFI), providing a bound to the precision, and multipartite entanglement quantifiers such as the entanglement depth. However, a connection between the QFI and entanglement monotones, i.e., functions that do not increase under Local Operations and Classical Communications, has so far remained elusive. In this Letter, we fill this gap by introducing a family of uncertainty relations that bound bipartite entanglement monotones from below via elements of a quantum Fisher information matrix. To further emphasize the significance of our results, we connect these relations to the achievable precision in multiparameter estimation. Considering a system split into two parts with arbitrary dimension, we also show that, while two-dimensional entanglement is sufficient to estimate a single parameter with maximal precision, genuine high-dimensional entanglement is required for multiparameter estimation. We conclude by illustrating how our method extends naturally to a multipartite splitting.
This post is the author's doctoral dissertation back in 1997. The dissertation covers following four kinds of problems: First, it studies achievable Cramer-Rao type bounds of various multi-parameter pure state models. Second, it relates CR-bouds, both of mixed state and pure states, with Berry-Uhlmann curvature. Roughly, it characterize how the statistical model differs from classical probability distribution family. Though the relation is rather qualitative for mixed states, the quantitative relation is obtained for pure state models. Third topic is relation between Berry-Uhlmann geometry and Amari-Nagaoka's quantum information geometry. Forth, various problems in quantum physics, uncertainty relations, measurement of temperature, time reversal symmetry, etc., are discussed using estimation theory.