Linear Gaussian Quantum State Smoothing: Understanding the optimal unravelings for Alice to estimate Bob's state

TL;DR

This paper advances the understanding of quantum state smoothing by deriving detailed equations for linear Gaussian states and proposing a hypothesis for Bob's optimal measurement strategy to maximize Alice's state purity improvement.

Contribution

It provides a detailed derivation of LGQ smoothing equations and introduces a hypothesis for Bob's optimal measurement strategy to enhance Alice's state estimation.

Findings

The hypothesis that Bob should observe the back-action from Alice's measurement is effective.

The derived equations clarify the role of Bob's measurement in quantum state smoothing.

The strategy improves the purity of Alice's estimated state compared to filtering alone.

Abstract

Quantum state smoothing is a technique to construct an estimate of the quantum state at a particular time, conditioned on a measurement record from both before and after that time. The technique assumes that an observer, Alice, monitors part of the environment of a quantum system and that the remaining part of the environment, unobserved by Alice, is measured by a secondary observer, Bob, who may have a choice in how he monitors it. The effect of Bob's measurement choice on the effectiveness of Alice's smoothing has been studied in a number of recent papers. Here we expand upon the Letter which introduced linear Gaussian quantum (LGQ) state smoothing [Phys. Rev. Lett., 122, 190402 (2019)]. In the current paper we provide a more detailed derivation of the LGQ smoothing equations and address an open question about Bob's optimal measurement strategy. Specifically, we develop a simple hypothesis that allows one to approximate the optimal measurement choice for Bob given Alice's measurement choice. By 'optimal choice' we mean the choice for Bob that will maximize the purity improvement of Alice's smoothed state compared to her filtered state (an estimated state based only on Alice's past measurement record). The hypothesis, that Bob should choose his measurement so that he observes the back-action on the system from Alice's measurement, seems contrary to one's intuition about quantum state smoothing. Nevertheless we show that it works even beyond a linear Gaussian setting.