Quantum state smoothing as an optimal estimation problem with three different cost functions

TL;DR

This paper demonstrates that quantum state smoothing provides an optimal estimate under certain cost functions, but not for all, and introduces a new optimal estimator called the lustrated smoothed state.

Contribution

It establishes the optimality of quantum state smoothing for specific cost functions and introduces the lustrated smoothed state as an optimal estimator for linear infidelity.

Findings

Smoothed quantum state minimizes the trace-square deviation and relative entropy to the true state.

The smoothed state is not optimal for linear infidelity, leading to the proposal of the lustrated smoothed state.

The lustrated smoothed state is a pure eigenstate of the smoothed state with the largest eigenvalue.

Abstract

Quantum state smoothing is a technique to estimate an unknown true state of an open quantum system based on partial measurement information both prior and posterior to the time of interest. In this paper, we show that the smoothed quantum state is an optimal state estimator; that is, it minimizes a risk (expected cost) function. Specifically, we show that the smoothed quantum state is optimal with respect to two cost functions: the trace-square deviation from and the relative entropy to the unknown true state. However, when we consider a related risk function, the linear infidelity, we find, contrary to what one might expect, that the smoothed state is not optimal. For this case, we derive the optimal state estimator, which we call the lustrated smoothed state. It is a pure state, the eigenstate of the smoothed quantum state with the largest eigenvalue.