# How to estimate past measurement interventions on a quantum system undergoing continuous monitoring

## TL;DR

This paper investigates the estimation of past quantum states in continuously monitored systems, demonstrating that such measurements are always commuting and formulating a consistent statistical inference framework for quantum smoothing.

## Contribution

It provides a rigorous analysis showing that measuring noncommuting observables in this context is not possible, and extends the formalism to multiple external measurements during continuous monitoring.

## Key findings

- Measurements are always commuting, respecting quantum non-demolition principles.
- A well-posed statistical inference problem for quantum smoothing is established.
- The formalism is extended to multiple external measurements over time.

## Abstract

We analyze the problem of estimating past quantum states of a monitored system from a mathematical perspective in order to ensure self-consistency with the principle of quantum non-demolition. Despite several claims of ``measuring noncommuting observables'' in the physics literature, we show that we are always measuring commuting processes. Our main interest is in the notion of quantum smoothing or retrodiction. In particular, we examine proposals to estimate the result of an external measurement made on an open quantum systems during a period where it is also undergoing continuous monitoring. A full analysis shows that the non-demolition principle is not actually violated, and so a well-posed as a statistical inference problem can be formulated. We extend the formalism to consider multiple independent external measurements made on the system over the course of a continual period of monitoring.