Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0904.1969 · Physical Review Letters

Time-Symmetric Quantum Theory of Smoothing

Mankei Tsang

Published Jun 25, 2009• 113 citations
arXivHTMLPDF

Abstract

Smoothing is an estimation technique that takes into account both past and future observations and can be more accurate than filtering alone. In this Letter, a quantum theory of smoothing is constructed using a time-symmetric formalism, thereby generalizing prior work on classical and quantum filtering, retrodiction, and smoothing. The proposed theory solves the important problem of optimally estimating classical Markov processes coupled to a quantum system under continuous measurements, and is thus expected to find major applications in future quantum sensing systems, such as gravitational wave detectors and atomic magnetometers.

DOI
10.1103/physrevlett.102.250403

Related papers

Time-Symmetric Quantum Smoothing: A General Theory of Optimal Quantum Sensing

Mankei Tsang

The important problem of optimal waveform estimation for quantum sensing is solved using a time-symmetric approach. The theory generalizes prior work in classical and quantum estimation and can significantly out-perform previously proposed techniques.

Conference on Lasers and Electro-Optics•Jan 1, 2010•0 citations

Adaptive Optical Phase Estimation Using Time-Symmetric Quantum Smoothing

Trevor A. Wheatley, Dominic William Berry, Hidehiro Yonezawa +7 more

Quantum parameter estimation has many applications, from gravitational wave detection to quantum key distribution. The most commonly used technique for this type of estimation is quantum filtering, using only past observations. We present the first experimental demonstration of quantum smoothing, a time-symmetric technique that uses past and future observations, for quantum parameter estimation. We consider both adaptive and nonadaptive quantum smoothing, and show that both are better than their filtered counterparts. For the problem of estimating a stochastically varying phase shift on a coherent beam, our theory predicts that adaptive quantum smoothing (the best scheme) gives an estimate with a mean-square error up to 2sqrt[2] times smaller than nonadaptive filtering (the standard quantum limit). The experimentally measured improvement is 2.24+/-0.14.

arXiv:0912.1162•Physical Review Letters•Mar 3, 2010•95 citations

Optimality of the Petz-Fuchs Smoothed State

Kiarn T. Laverick

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.

quant-pharXiv:2610.41311•Nov 2, 2026

Unifying quantum smoothing theories with extended retrodiction

Mingxuan Liu, Ge Bai, Valerio Scarani

Estimating the state of an open quantum system monitored over time requires incorporating information from past measurements (filtering) and, for improved accuracy, also from future measurements (smoothing). While classical smoothing is well understood within a Bayesian framework, its quantum generalization has been challenging, leading to distinct and seemingly incompatible approaches. In this work, we demonstrate that quantum state smoothing hinges on a uniquely quantum feature: the fundamental dependence of retrodiction on prior correlations. We introduce auxiliary systems into the prior to represent specified correlations formed during preparation and evolution, and formulate quantum state smoothing as extended Bayesian retrodiction. This framework identifies previous approaches as different choices of the extended prior, and naturally extends it to other choices that have not been considered before. We also give an information-theoretic characterization of the choices of prior, in terms of the average entropy of the smoothed states. Our results establish quantum state smoothing as a fundamentally retrodictive process just like classical smoothing, with proper quantum features clearly identified.

arXiv:2510.08447•npj Quantum Information•Nov 1, 2026•0 citations

Retrodictive approach to quantum state smoothing

Mingxuan Liu, Valerio Scarani, Alexia Auffèves +1 more

Smoothing is a technique for estimating the state of an imperfectly monitored open system by combining both prior and posterior measurement information. In the quantum regime, current approaches to smoothing either give unphysical outcomes, due to the non-commutativity of the measurements at different times, or require assumptions about how the environment is measuring the system, which with current technology is unverifiable. We propose a novel definition of the smoothed quantum state based on quantum Bayesian retrodiction, which mirrors the classical retrodictive approach to smoothing. This approach always yields physical results and does not require any assumption on the environment. We show that this smoothed state has, on average, greater purity than the state reconstructed using just the prior information. Finally, we make a connection with the smoothing theory of Guevara and Wiseman in a well-studied regime, and describe from a purely quantum perspective how it conditions on the posterior information.

arXiv:2501.15986•Physical Review A•Sep 2, 2025•1 citation

Quantum State Smoothing Cannot Be Assumed Classical Even When the Filtering and Retrofiltering Are Classical

Kiarn T. Laverick, Prahlad Warszawski, Areeya Chantasri +1 more

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.

arXiv:2305.12970•PRX Quantum•Dec 11, 2023•3 citations
Smoothing is an estimation technique that takes into account both past and future observations and can be more accurate than filtering alone. In this Letter, a quantum theory of smoothing is constructed using a time-symmetric formalism, thereby generalizing prior work on classical and quantum filtering, retrodiction, and smoothing. The proposed theory solves the important problem of optimally estimating classical Markov processes coupled to a quantum system under continuous measurements, and is thus expected to find major applications in future quantum sensing systems, such as gravitational wave detectors and atomic magnetometers.
Quantum parameter estimation has many applications, from gravitational wave detection to quantum key distribution. The most commonly used technique for this type of estimation is quantum filtering, using only past observations. We present the first experimental demonstration of quantum smoothing, a time-symmetric technique that uses past and future observations, for quantum parameter estimation. We consider both adaptive and nonadaptive quantum smoothing, and show that both are better than their filtered counterparts. For the problem of estimating a stochastically varying phase shift on a coherent beam, our theory predicts that adaptive quantum smoothing (the best scheme) gives an estimate with a mean-square error up to 2sqrt[2] times smaller than nonadaptive filtering (the standard quantum limit). The experimentally measured improvement is 2.24+/-0.14.
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.
Estimating the state of an open quantum system monitored over time requires incorporating information from past measurements (filtering) and, for improved accuracy, also from future measurements (smoothing). While classical smoothing is well understood within a Bayesian framework, its quantum generalization has been challenging, leading to distinct and seemingly incompatible approaches. In this work, we demonstrate that quantum state smoothing hinges on a uniquely quantum feature: the fundamental dependence of retrodiction on prior correlations. We introduce auxiliary systems into the prior to represent specified correlations formed during preparation and evolution, and formulate quantum state smoothing as extended Bayesian retrodiction. This framework identifies previous approaches as different choices of the extended prior, and naturally extends it to other choices that have not been considered before. We also give an information-theoretic characterization of the choices of prior, in terms of the average entropy of the smoothed states. Our results establish quantum state smoothing as a fundamentally retrodictive process just like classical smoothing, with proper quantum features clearly identified.
Smoothing is a technique for estimating the state of an imperfectly monitored open system by combining both prior and posterior measurement information. In the quantum regime, current approaches to smoothing either give unphysical outcomes, due to the non-commutativity of the measurements at different times, or require assumptions about how the environment is measuring the system, which with current technology is unverifiable. We propose a novel definition of the smoothed quantum state based on quantum Bayesian retrodiction, which mirrors the classical retrodictive approach to smoothing. This approach always yields physical results and does not require any assumption on the environment. We show that this smoothed state has, on average, greater purity than the state reconstructed using just the prior information. Finally, we make a connection with the smoothing theory of Guevara and Wiseman in a well-studied regime, and describe from a purely quantum perspective how it conditions on the posterior information.
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.