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arXiv:1505.06437 · Journal of Mathematical Physics

Explicit formula for the Holevo bound for two-parameter qubit-state estimation problem

Jun Suzuki

Published Apr 1, 2016• 75 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/1.4945086

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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.
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.
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.
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.
In quantum estimation theory, the Holevo bound is known as a lower bound of weighted traces of covariances of unbiased estimators. The Holevo bound is defined by a solution of a minimization problem, and in general, an explicit solution is not known. When the dimension of Hilbert space is 2 and the number of parameters is 2, an explicit form of the Holevo bound was given by Suzuki. In this paper, we focus on a logarithmic derivative that lies between the symmetric logarithmic derivative (SLD) and the right logarithmic derivative parameterized by β ∈ [0, 1] to obtain lower bounds of the weighted trace of covariance of an unbiased estimator. We introduce the maximum logarithmic derivative bound as the maximum of bounds with respect to β. We show that all monotone metrics induce lower bounds, and the maximum logarithmic derivative bound is the largest bound among them. We show that the maximum logarithmic derivative bound has explicit solution when the d dimensional model has d + 1 dimensional D invariant extension of the SLD tangent space. Furthermore, when d = 2, we show that the maximization problem to define the maximum logarithmic derivative bound is the Lagrangian duality of the minimization problem to define the Holevo bound and is the same as the Holevo bound. This explicit solution is a generalization of the solution for a two-dimensional Hilbert space given by Suzuki. We also give examples of families of quantum states to which our theory can be applied not only for two-dimensional Hilbert spaces.
Abstract Multiparameter quantum estimation theory aims to determine simultaneously the ultimate precision of all parameters contained in the state of a given quantum system. Determining this ultimate precision depends on the quantum Fisher information matrix (QFIM) which is essential to obtaining the quantum Cramér–Rao bound. This is the main motivation of this work which concerns the computation of the analytical expression of the QFIM. Inspired by the results reported in J. Phys. A: Math. Theor. 52 035304 (2019), the general formalism of the multiparameter quantum estimation theory of quantum Gaussian states in terms of their first and second moments is given. We give the analytical formulas of right logarithmic derivative and symmetric logarithmic derivative operators. Then we derive the general expressions of the corresponding quantum Fisher information matrices. We also derive an explicit expression of the condition which ensures the saturation of the quantum Cramér–Rao bound in estimating several parameters. Finally, we examine some examples to clarify the use of our results.
We investigate the quantum Cramer-Rao bounds on the joint multiple-parameter estimation with the Gaussian state as a probe. We derive the explicit right logarithmic derivative and symmetric logarithmic derivative operators in such a situation. We compute the corresponding quantum Fisher information matrices, and find that they can be fully expressed in terms of the mean displacement and covariance matrix of the Gaussian state. Finally, we give some examples to show the utility of our analytical results.