Efficient computation of the Nagaoka—Hayashi bound for multi-parameter estimation with separable measurements

Abstract

Efficient computation of theNagaoka-Hayashi bound for multiparameter estimation with separable measurements.

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 [26]. Consider the Hilbert space H := C n ⊗ H , and define L on H whose ( j, k) block -matrix 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. [26]. Combining Lemma 1 and Lemma 2 gives the main result of the paper.…”

“…The main difference from the Nagaoka-Hayashi bound for the point estimation setting [26] is that there is no constraint about the locally unbiasedness. The next result shows that the proposed Bayesian Nagaoka-Hayashi bound can be computed efficiently by a semidefinite programming (SDP) problem.…”

“…Other constraints on L and X can also be put in the trace condition for the variable (see Supplementary Note 4 of Ref. [26] how to implement these conditions as semidefinite programming).…”

“…This Nagaoka's result was generalized by Hayashi for any finite number of noncommuting matrices [25]. In a recent paper [26], 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.…”