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arXiv:2506.15951 · New Journal of Physics

Quantum state smoothing when Alice assumes the wrong type of monitoring by Bob

Areeya Chantasri, Kiarn T. Laverick, Howard M. Wiseman

Published Oct 27, 2025• 0 citations
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Abstract

Abstract An open quantum system leaks information into its environment. We are interested in physical situations where an observer, say Alice, can recover some of that information, as a classical measurement record. The optimal way for Alice to estimate the quantum state at time t from the record before t is known as quantum filtering. Recently, a version of quantum smoothing, in which Alice estimates the state at time t using her record on both sides of t , has been developed. It requires Alice to make optimal inferences about the pre- t record of a second observer, say Bob, who recovers whatever information Alice does not. But for Alice to make this inference, she needs to know Bob’s measurement setup. In this paper we consider what happens if Alice is mistaken in her assumption about Bob’s setup. We show that the accuracy—as measured by the trace-squared-deviation, of Alice’s estimate of the true state (i.e. the state conditioned on her and Bob’s pre- t records)—depends strongly on her setup, Bob’s actual setup, and the wrongly assumed setup. Using resonance fluorescence as a model system, we show numerically that in some cases the wrong smoothing is almost as accurate as the right smoothing, but in other cases much less accurate, even being less accurate than Alice’s filtered estimate. Curiously, in some of the latter cases the fidelity of Alice’s wrong estimate with the true state is actually higher than that of her right estimate. We explain this, and other features we observe numerically, by some simple analytical arguments.

DOI
10.1088/1367-2630/ae17e3

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Abstract An open quantum system leaks information into its environment. We are interested in physical situations where an observer, say Alice, can recover some of that information, as a classical measurement record. The optimal way for Alice to estimate the quantum state at time t from the record before t is known as quantum filtering. Recently, a version of quantum smoothing, in which Alice estimates the state at time t using her record on both sides of t , has been developed. It requires Alice to make optimal inferences about the pre- t record of a second observer, say Bob, who recovers whatever information Alice does not. But for Alice to make this inference, she needs to know Bob’s measurement setup. In this paper we consider what happens if Alice is mistaken in her assumption about Bob’s setup. We show that the accuracy—as measured by the trace-squared-deviation, of Alice’s estimate of the true state (i.e. the state conditioned on her and Bob’s pre- t records)—depends strongly on her setup, Bob’s actual setup, and the wrongly assumed setup. Using resonance fluorescence as a model system, we show numerically that in some cases the wrong smoothing is almost as accurate as the right smoothing, but in other cases much less accurate, even being less accurate than Alice’s filtered estimate. Curiously, in some of the latter cases the fidelity of Alice’s wrong estimate with the true state is actually higher than that of her right estimate. We explain this, and other features we observe numerically, by some simple analytical arguments.
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.
Abstract Tracking the dynamics of a quantum system is conventionally achieved by monitoring the system continuously in time and filtering the information contained in measurement records via the causal quantum trajectory approach. However, in practical scenarios, there is often loss of information to the environment, leading to filtered states that are impure because of decoherence. If real-time tracking is not required, the lost information can be maximally extracted via acausal quantum state smoothing , which has been theoretically proven to better restore the system’s coherence (purity) than causal filtering. Interestingly, quantum state smoothing requires assumptions about how any lost quantum information (unobserved by the experimenter) was turned into classical information by the environment. In this work, we experimentally demonstrate smoothing scenarios, using an optical parametric oscillator and introducing ‘observed’ and ‘unobserved’ channels by splitting the output beam into two independent homodyne detectors. We achieve improvement in state purification of 10.3% ± 1.6%, squeezing restoration of 7.6% ± 2.6%, and show that smoothed states are better estimates of hidden true states than those from conventional filtering. The estimation techniques used in this paper are promising for many applications in quantum information that incorporate post-processing.
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.
Tracking the dynamics of a quantum system is conventionally achieved by monitoring the system continuously in time and filtering the information contained in measurement records via the causal quantum trajectory approach. However, in practical scenarios there is often loss of information to the environment, leading to filtered states that are impure because of decoherence. If real-time tracking is not required, the lost information can be maximally extracted via acausal quantum state smoothing, which has been theoretically proven to better restore the system's coherence (purity) than causal filtering. Interestingly, quantum state smoothing requires assumptions of how any lost quantum information (unobserved by the experimenter) was turned into classical information by the environment. In this work, we experimentally demonstrate smoothing scenarios, using an optical parametric oscillator and introducing `observed' and `unobserved' channels by splitting the output beam into two independent homodyne detectors. We achieve improvement in state purification of 10.3% +/- 1.6%, squeezing restoration of 7.6% +/- 2.6%, and show that smoothed states are better estimates of hidden true states than those from conventional filtering. The estimation techniques used in this paper are promising for many applications in quantum information that incorporate post-processing.
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.
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.