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Proceedings of the ISCIE International Symposium on Stochastic Systems Theory and its Applications

On Recursive Quantum State Smoothing

Kentaro Ohki

Published Mar 31, 2022• 0 citations
Publisher

Abstract

Past quantum state estimation is a controversial issue, and it has been studied from both theoretical curiosity and practical necessity. Some studies propose quantum state smoothing methods, yet it is still an open problem in what situations ensure the past quantum states. This paper discusses conditions that the past quantum observable estimation becomes a past quantum state estimation and proposes two recursive methods of quantum state smoother.

DOI
10.5687/sss.2022.8

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Past quantum state estimation is a controversial issue, and it has been studied from both theoretical curiosity and practical necessity. Some studies propose quantum state smoothing methods, yet it is still an open problem in what situations ensure the past quantum states. This paper discusses conditions that the past quantum observable estimation becomes a past quantum state estimation and proposes two recursive methods of quantum state smoother.
Quantum state smoothing is an estimation technique that uses past and future measurement information obtained from a continuous-in-time measurement record to reconstruct the state of an open quantum system. Recently, there has been a new approach to quantum state smoothing based on quantum retrodiction known as the Petz-Fuchs smoothed state. Here I prove that this Petz-Fuchs smoothed state majorizes all other smoothed states that only depend on the filtered state (the operator conditioned on the past measurement record) and the retrofiltered effect (the operator describing the future measurement record). To do this, I characterize the set of all possible smoothed states satisfying this condition.
State smoothing is a technique to estimate a state at a particular time, conditioned on information obtained both before (past) and after (future) that time.For a classical system, the smoothed state is a normalized product of the filtered state (a state conditioned only on the past measurement information and the initial preparation) and the retrofiltered effect (depending only on the future measurement information).For the quantum case, while there are well-established analogues of the filtered state (ρ F ) and the retrofiltered effect ( ÊR ), their product does not, in general, provide a valid quantum state for smoothing.However, this procedure does seem to work when ρ F and ÊR are mutually diagonalizable.This fact has been used to obtain smoothed quantum states-purer than the filtered states-in a number of experiments on continuously monitored quantum systems, in cavity QED and atomic systems.In this paper we show that there is an implicit assumption underlying this technique: that if all the information were known to the observer, the true system state would be one of the diagonal basis states.This assumption does not necessarily hold, as the missing information is quantum information.It could be known to the observer only if it were turned into a classical measurement record, but then its nature would depend on the choice of measurement.We show by a simple model that, depending on that measurement choice, the smoothed quantum state can: agree with that from the classical method, disagree with it but still be co-diagonal with it, or not even be co-diagonal with it.That is, just because filtering and retrofiltering appear classical does not mean classical smoothing theory is applicable in quantum experiments.
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.
Smoothing is an estimation method whereby a classical state (probability distribution for classical variables) at a given time is conditioned on all-time (both earlier and later) observations. Here we define a smoothed quantum state for a partially monitored open quantum system, conditioned on an all-time monitoring-derived record. We calculate the smoothed distribution for a hypothetical unobserved record which, when added to the real record, would complete the monitoring, yielding a pure-state "quantum trajectory." Averaging the pure state over this smoothed distribution yields the (mixed) smoothed quantum state. We study how the choice of actual unraveling affects the purity increase over that of the conventional (filtered) state conditioned only on the past record.
Abstract We investigate the estimation technique called quantum state smoothing introduced in (Guevara and Wiseman 2015 Phys. Rev. Lett. 115 180407), 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. The technique was shown to give a better estimate of the underlying true quantum states 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.