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Chinese Physics B

Holevo bound independent of weight matrices for estimating two parameters of a qubit

Chang Niu, Sixia Yu

Published Dec 1, 2023• 1 citation
Publisher

Abstract

Holevo bound plays an important role in quantum metrology as it sets the ultimate limit for multi-parameter estimations, which can be asymptotically achieved. Except for some trivial cases, the Holevo bound is implicitly defined and formulated with the help of weight matrices. Here we report the first instance of an intrinsic Holevo bound, namely, without any reference to weight matrices, in a nontrivial case. Specifically, we prove that the Holevo bound for estimating two parameters of a qubit is equivalent to the joint constraint imposed by two quantum Cramér–Rao bounds corresponding to symmetric and right logarithmic derivatives. This weightless form of Holevo bound enables us to determine the precise range of independent entries of the mean-square error matrix, i.e., two variances and one covariance that quantify the precisions of the estimation, as illustrated by different estimation models. Our result sheds some new light on the relations between the Holevo bound and quantum Cramér–Rao bounds. Possible generalizations are discussed.

DOI
10.1088/1674-1056/ad117d

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Holevo bound plays an important role in quantum metrology as it sets the ultimate limit for multi-parameter estimations, which can be asymptotically achieved. Except for some trivial cases, the Holevo bound is implicitly defined and formulated with the help of weight matrices. Here we report the first instance of an intrinsic Holevo bound, namely, without any reference to weight matrices, in a nontrivial case. Specifically, we prove that the Holevo bound for estimating two parameters of a qubit is equivalent to the joint constraint imposed by two quantum Cramér–Rao bounds corresponding to symmetric and right logarithmic derivatives. This weightless form of Holevo bound enables us to determine the precise range of independent entries of the mean-square error matrix, i.e., two variances and one covariance that quantify the precisions of the estimation, as illustrated by different estimation models. Our result sheds some new light on the relations between the Holevo bound and quantum Cramér–Rao bounds. Possible generalizations are discussed.
The quantum Cramér-Rao bound constrains the precision of parameter estimation through the symmetric logarithmic derivative quantum Fisher information (SLD QFI). It is asymptotically achievable for regular single-parameter estimation models, but generally not in the multiparameter setting, where optimal measurements for different parameters may be incompatible. For multiparameter estimation, allowing collective measurements, the corresponding asymptotically achievable precision limit is the Holevo bound, whose standard formulation is an optimization over auxiliary Hermitian operators. We bridge this apparent difference in formulation by giving an exact characterization of the Holevo bound in terms of a family of QFIs interpolating between the SLD and right logarithmic derivative (RLD) QFIs. Specifically, for any locally identifiable finite-dimensional estimation problem, we prove that the operator-feasible region in the standard variational definition of the Holevo bound coincides with the intersection of the regions defined by the corresponding Cramér-Rao-type matrix constraints for the entire QFI family. This characterization also applies to rank-deficient states and does not require any prior choice of weight matrix. Consequently, the conventional weight-dependent Holevo bound is recovered by minimizing the corresponding weighted cost over the resulting common feasible region.
The main contribution of this paper is to derive an explicit expression for the fundamental precision bound, the Holevo bound, for estimating any two-parameter family of qubit mixed-states in terms of quantum versions of Fisher information. The obtained formula depends solely on the symmetric logarithmic derivative (SLD), the right logarithmic derivative (RLD) Fisher information, and a given weight matrix. This result immediately provides necessary and sufficient conditions for the following two important classes of quantum statistical models; the Holevo bound coincides with the SLD Cramér-Rao bound and it does with the RLD Cramér-Rao bound. One of the important results of this paper is that a general model other than these two special cases exhibits an unexpected property: the structure of the Holevo bound changes smoothly when the weight matrix varies. In particular, it always coincides with the RLD Cramér-Rao bound for a certain choice of the weight matrix. Several examples illustrate these findings.
Multiparameter quantum estimation faces a fundamental challenge due to the inherent incompatibility of optimal measurements for different parameters, a direct consequence of quantum non-commutativity. This incompatibility is quantified by the gap between the symmetric logarithmic derivative (SLD) quantum Cramér-Rao bound, which is not always attainable, and the asymptotically achievable Holevo bound. This work provides a comprehensive analysis of this gap by introducing and contrasting two scalar measures. The first is the weight-independent quantumness measure RRR, which captures the intrinsic incompatibility of the estimation model. The second is a tighter, weight-dependent measure T[W]T[W]T[W] which explicitly incorporates the cost matrix WWW assigning relative importance to different parameters. We establish a hierarchy of bounds based on these two measures and derive necessary and sufficient conditions for their saturation. Through analytical and numerical studies of tunable qubit and qutrit models with SU(2) unitary encoding, we demonstrate that the weight-dependent bound CT[W]C_{T}[W]CT​[W] often provides a significantly tighter approximation to the Holevo bound CH[W]C_{H}[W]CH​[W] than the RRR-dependent bound, especially in higher-dimensional systems. We also develop an approach based on CT[W]C_{T}[W]CT​[W] to compute the Holevo bound CH[W]C_{H}[W]CH​[W] analytically. Our results highlight the critical role of the weight matrix's structure in determining the precision limits of multiparameter quantum metrology.
The estimation of more than one parameter in quantum mechanics is a fundamental problem with relevant practical applications. In fact, the ultimate limits in the achievable estimation precision are ultimately linked with the non-commutativity of different observables, a peculiar property of quantum mechanics. We here consider several estimation problems for qubit systems and evaluate the corresponding quantumnessR, a measure that has been recently introduced in order to quantify how incompatible the parameters to be estimated are. In particular, R is an upper bound for the renormalized difference between the (asymptotically achievable) Holevo bound and the SLD Cramér-Rao bound (i.e., the matrix generalization of the single-parameter quantum Cramér-Rao bound). For all the estimation problems considered, we evaluate the quantumness R and, in order to better understand its usefulness in characterizing a multiparameter quantum statistical model, we compare it with the renormalized difference between the Holevo and the SLD-bound. Our results give evidence that R is a useful quantity to characterize multiparameter estimation problems, as for several quantum statistical model, it is equal to the difference between the bounds and, in general, their behavior qualitatively coincide. On the other hand, we also find evidence that, for certain quantum statistical models, the bound is not in tight, and thus R may overestimate the degree of quantum incompatibility between parameters.
We discuss a problem of parameter estimation for quantum two-level system, qubit system, in presence of unknown phase parameter. We analyze trade-off relations for mean square errors (MSEs) when estimating relevant parameters with separable measurements based on known precision bounds; the symmetric logarithmic derivative (SLD) Cramér–Rao (CR) bound and Hayashi–Gill–Massar (HGM) bound. We investigate the optimal measurement which attains the HGM bound and discuss its properties. We show that the HGM bound for relevant parameters can be attained asymptotically by using some fraction of given n quantum states to estimate the phase parameter. We also discuss the Holevo bound which can be attained asymptotically by a collective measurement.
Abstract Finding the optimal attainable precisions in quantum multiparameter metrology is a non-trivial problem. One approach to tackling this problem involves the computation of bounds which impose limits on how accurately we can estimate certain physical quantities. One such bound is the Holevo Cramér–Rao bound on the trace of the mean squared error matrix. The Holevo bound is an asymptotically achievable bound when one allows for any measurement strategy, including collective measurements on many copies of the probe. In this work, we introduce a tighter bound for estimating multiple parameters simultaneously when performing separable measurements on a finite number of copies of the probe. This makes it more relevant in terms of experimental accessibility. We show that this bound can be efficiently computed by casting it as a semidefinite programme. We illustrate our bound with several examples of collective measurements on finite copies of the probe. These results have implications for the necessary requirements to saturate the Holevo bound.