Efficient computation of the Nagaoka–Hayashi bound for multiparameter estimation with separable measurements

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.

Approaching optimal entangling collective measurements on quantum computing platforms

“…In Fig. 2f, we compute the Nagaoka bound for performing collective measurements on up to seven copies of the probe state simultaneously, corresponding to a 128-dimensional Hilbert space 40 .…”

Bayesian Nagaoka-Hayashi Bound for Multiparameter Quantum-State Estimation Problem

“…We next define a matrix and a vector which are defined on the extended Hilbert space [25]. Consider the Hilbert space H := C n ⊗ H, and define L on H whose jk component is given by L jk .…”

“…To derive a lower bound for the Bayes risk R[Π, θ], we follow the same line of logic used in Ref. [25]. This then gives the main result of the paper.…”

“…This Nagaoka's result was generalized by Hayashi for any finite number of noncommuting matrices [24]. In a recent paper [25], the Nagaoka bound for parameter estimation was generalized to estimating any number of non-random parameters, which was named as the Nagaoka-Hayashi bound. In this paper, we attempt to make a further step toward developing genuine quantum bounds based on the Nagaoka-Hayashi bound in Bayesian parameter estimation.…”

Optimal estimation of conjugate shifts in position and momentum by classically correlated probes and measurements

“…For any particular measurement strategy, these variances are bounded by the inverse of the classical Fisher information (CFI) Their ultimate limit is given by the Holevo-Cramer-Rao (HCR) bound [3,11] that can be obtained by minimization over all possible measurement strategies, which can, in many cases, be only done numerically. Numerical computation can be also used to obtain the Nagaoka-Hayashi bound for separable single copy measurements [29,30]. The variances are also lower bounded by the inverse of the quantum Fisher information (QFI) obtained either from symmetric or right logarithmic derivative [3,6,12], but this bound is not always tight for multiparameter quantum estimation.…”