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Physical Review A

Anisotropic efficiency of adaptive measurements in two-parameter estimation for states in manifolds with nonzero Berry curvature

Haijun Xing

Published Jul 22, 2024• 1 citation
Publisher

Abstract

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.

DOI
10.1103/physreva.110.012449

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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.
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.
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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.
The geometry of quantum states has profound implications in quantum multiparameter estimation. While the Riemannian structure of quantum state space is well understood, the full understanding of the curvature structure of mixed quantum states is still an open problem. Inspired by the Yang-Mills action in non-Abelian gauge theory, we propose a scalar quantifying the Uhlmann curvature and establish its connection to the measurement incompatibility in quantum multiparameter estimation problems. We show that this curvature measure is gauge invariant, reparametrization invariant, and vanishes if and only if the Uhlmann curvature vanishes. We also explicitly calculate the Uhlmann curvature for the joint estimation of phase and phase diffusion as an example.
We study the local efficiency of symmetric informationally complete positive operator-valued measures (SIC-POVMs) from an information geometrical perspective. The local efficiency of a SIC-POVM is defined as the relative magnitude of the density of distinguishable states acquired via the measurement over the intrinsic density of quantum states. The two densities measure the local distinguishability of quantum states and its corresponding quantum resources, respectively. Strikingly, the efficiency is intrinsic in the sense of invariance under reparametrization. Specific to two-level systems, the efficiency is highly anisotropic and has a tetrahedral symmetry. Especially for a shell of the Bloch sphere, states in directions parallel (antiparallel) to the measurement basis have minimal (maximal) efficiency. Furthermore, the optimality of SIC-POVM versus other equal-length measurements is shown.
Multiparameter quantum estimation faces two fundamental obstacles: sloppiness, i.e., anisotropy of the quantum Fisher information matrix (QFIM) that renders some parameter directions insensitive, and incompatibility, the non-commutativity of optimal measurements for different parameters. The trade-off bound CTC_TCT​ captures their joint impact on precision, but it has remained unclear how the distribution of incompatibility across parameter planes affects its overall cost. Here we separate the total amount of incompatibility from its location. We introduce a dimensionless quantity Gn(F)G_n^{(F)}Gn(F)​ that measures the alignment between the incompatibility distribution and the eigenvalues of the QFIM, and show how the Frobenius scale of the incompatibility contribution factorizes. We obtain a bound and prove the incompatibility cost lies between this bound and a rank-dependent multiple thereof. We also prove that at fixed sloppiness, or equivalently fixed Fisher volume, concentrating incompatibility into a single parameter plane reduces the optimized trade-off cost because the Fisher geometry can then be reshaped to allocate more Fisher area to that plane. A qutrit SU(2)SU(2)SU(2) encoding numerically confirms that states with larger incompatibility strength can nevertheless incur a smaller cost if the matching factor GGG is sufficiently small. Our results establish that the distribution of incompatibility relative to the Fisher eigenbasis is a central diagnostic for multiparameter estimation, beyond the total incompatibility strength.