Quantum estimation for non-differentiable models

Abstract

State estimation is a classical problem in quantum information. In optimization of estimation scheme, to find a lower bound to the error of the estimator is a very important step. So far, all the proposed tractable lower bounds use derivative of density matrix. However, sometimes, we are interested in quantities with singularity, e.g. concurrence etc. In the paper, lower bounds to a Mean Square Error (MSE) of an estimator are derived for a quantum estimation problem without smoothness assumptions. Our main idea is to replace the derivative by difference, as is done in classical estimation theory. We applied the inequalities to several examples, and derived optimal estimator for some of them.

Precision measurements of temperature and chemical potential of quantum gases

“…The state consequently changes into an orthogonal state with a different number of particles. This non-smooth change is in contrast with the assumptions upon which the standard quantum Cramér-Rao bound (5) is based, and requires the generalization to non-differentiable models [59]. However, this gives an intuition for the divergence of (F…”

“…Quantum estimation theory for non-differentiable models [59] needs to be applied when the temperature is exactly zero, as discussed in the appendix B.…”

“…If this assumption is not met, the quantum Cramér-Rao bound is generalized by the quantum Hammerseley-Chapman-Robbins-Kshirsagar bound [59]. The Fermi-Dirac distribution involved in the grand canonical thermal state (32) is the step function at zero temperature.…”

Attaining the Ultimate Precision Limit in Quantum State Estimation

“…The PDF of ℘ t0,t|m is upper bounded by J t0 . When the PDF of ℘ t0,t|m is differentiable with respect to t, equality in (25) holds if and only if ℘ t0,t|m is the normal distribution with average zero and variance J −1 t0 .…”

“…In particular, no previous study showed the validity of the Holevo bound under the weaker condition of local asymptotic covariance in the multiparameter scenario. To avoid employing the (local) unbiasedness condition, we focus on the discretized version of the RLD Fisher information matrix, introduced by Tsuda and Matsumoto [25]. Using this version of the RLD Fisher information matrix, we manage to handle the local asymptotic covariance condition and to show the validity of the Holevo bound in this broader scenario.…”