# Quantum state smoothing: Why the types of observed and unobserved measurements matter

## TL;DR

This paper analyzes how the types of observed and unobserved measurements affect the effectiveness of quantum state smoothing in estimating true quantum states, providing systematic insights and predictive methods for different detection scenarios.

## Contribution

It systematically investigates the impact of measurement types on quantum state smoothing power and introduces a correlation-based method to predict smoothing effectiveness.

## Key findings

- Smoothing power varies significantly with measurement types.
- Correlation strength between detection records predicts smoothing performance.
- The proposed prediction method aligns well with numerical simulations.

## Abstract

We investigate the estimation technique called quantum state smoothing introduced by Guevara and Wiseman [Phys.~Rev.~Lett.~{\bf 115}, 180407 (2015)], which offers a valid quantum state estimate for a partially monitored system, conditioned on the observed record both prior and posterior to an estimation time. Partial monitoring by an observer implies that there may exist records unobserved by that observer. It was shown that, given only the observed record, the observer can better estimate the underlying true quantum states, by inferring the unobserved record and using quantum state smoothing, rather than the usual quantum filtering approach. However, the improvement in estimation fidelity, originally examined for a resonantly driven qubit coupled to two vacuum baths, was also shown to vary depending on the types of detection used for the qubit's fluorescence. In this work, we analyse this variation in a systematic way for the first time. We first define smoothing power using an average purity recovery and a relative average purity recovery, of smoothing over filtering. Then, we explore the power for various combinations of fluorescence detection for both observed and unobserved channels. We next propose a method to explain the variation of the smoothing power, based on multi-time correlation strength between fluorescence detection records. The method gives a prediction of smoothing power for different combinations, which is remarkably successful in comparison with numerically simulated qubit trajectories.