When an aircraft or self-driving car determines its position, the sensor readings always contain noise. Estimating the current position in real time from data collected while moving is called "filtering." By contrast, "smoothing" is the process of re-estimating the position at the moment a vehicle passed a given point, using observation records from both before and after that moment, once all the travel data has been recorded. Because information obtained at later times is also used, the influence of random sensor errors can be suppressed and the past trajectory can be reconstructed more accurately. This is a technique widely used in classical signal processing.
However, the idea of using later observation records to precisely estimate a past state cannot be applied directly to quantum systems. In quantum mechanics, the act of measurement itself affects the state of the object being measured. If classical probability calculations are applied naively, the results can be physically impermissible: fluctuations in position and momentum may become small enough to violate Heisenberg's uncertainty principle at the same time, or values may appear that cannot be interpreted as probability distributions.
An international collaboration led by Shoryu Yokoyama (senior research scientist) and Hidehiro Yonezawa (team director) of the Optical Quantum Control Research Team at the RIKEN Center for Quantum Computing, together with Warit Asavanant (assistant professor at the time of the research) of the University of Tokyo's Graduate School of Engineering and others, built an experimental system using light leaking from an optical cavity. They succeeded in demonstrating "quantum state smoothing," which estimates a past quantum state while satisfying the principles of quantum mechanics. The results were published online on October 21, 2026, in the journal npj Quantum Information.
The "future observations" used in quantum state smoothing do not reverse causality or predict phenomena that have yet to occur. It is a post-hoc analysis: after all measurements have finished, it refers to information in the stored time-series data that was recorded after the time being estimated. The technique does not change the past quantum state itself; it increases the information an observer has about that state and thereby improves estimation accuracy.
A theory for estimating past states without contradicting quantum mechanics
The difficulty of quantum state smoothing lies in the nature of measurement in quantum mechanics. To estimate the quantum state at time $t$, ordinary quantum filtering updates the state sequentially using observation signals obtained from the start of the measurement up to time $t$. Because information accumulates along the direction of causality, an estimate based on an appropriate quantum model satisfies the conditions required of a physical density matrix, such as positivity and unit trace.
Smoothing, which estimates the state at time $t$ using data obtained up to a later time $T$, has been used in classical control theory for more than half a century. In quantum systems, however, one must also account for the back-action of measurements made between $t$ and $T$ and for the dissipation of information into the environment. Naively applying classical Bayesian estimation can lead to states that cannot actually exist, because they break the uncertainty relations of quantum mechanics.
The theory of "quantum state smoothing" was proposed in 2015 to address this problem by Professor Howard M. Wiseman and Dr. Ivonne A. Guevara of Griffith University in Australia (DOI: 10.1103/PhysRevLett.115.180407).
Wiseman and colleagues focused on the fact that information the estimating observer cannot obtain flows out into the surrounding environment. They introduced a theoretical model in which an observer, "Alice," cannot access the environmental information, while another observer, "Bob," measures all of it. Using her own observation records from before and after time $t$, Alice statistically estimates what measurement results Bob was likely to have obtained. She then averages the physically valid quantum states corresponding to each of Bob's possible results, weighted by the estimated probabilities.
Because each state being averaged satisfies the conditions of quantum mechanics, the estimated state obtained by combining them probabilistically is also physically valid. In other words, the method can improve the estimate of a past state using later observation records while avoiding unphysical solutions that would violate the uncertainty principle. The 2015 work laid the theoretical foundation.
To verify estimation accuracy experimentally, however, a "true state" is needed as a point of comparison. This is the reference describing what state the quantum system was actually in at that time. In ordinary quantum experiments, it is difficult to fully capture the information lost to the environment. Directly verifying whether a smoothed estimate is closer to the true state than a conventional filtered estimate therefore remained a challenge.
Two detectors reproduce the "hidden observer"
To verify the accuracy of quantum state smoothing, the research group set up two observation paths in an optical system, corresponding to Alice and Bob. This made it possible to record, with a separate detector, information that is normally hidden from the observer and to use it to evaluate the estimates.
The experimental subject was "squeezed light" generated inside an optical cavity. A squeezed state is one in which the quantum fluctuation of one of the two mutually orthogonal quadrature components of the light's electric field is suppressed below the fluctuation of the vacuum state. These components are related in the same way as position $x$ and momentum $p$ in classical mechanics, and by the uncertainty principle, suppressing the fluctuation of one increases that of the other. Squeezed light plays an important role in high-precision quantum measurement and optical quantum computation.
The team split the light leaking out of the optical cavity into two paths with a beam splitter. A homodyne detector corresponding to Alice was placed on one path, and another homodyne detector corresponding to Bob on the other. Homodyne detection is a technique that measures a specific quadrature component by interfering a reference light with the light under measurement. With this configuration, observation information invisible to Alice was recorded on Bob's side, reproducing the two observers assumed in the theory within the laboratory.
| Method / role | Observation data used | Role in the experiment | Notes |
|---|---|---|---|
| Conventional filtering | Alice's past records (0 to $t$) | Baseline mimicking real-time estimation | Does not use observation data after time $t$ |
| Quantum state smoothing | Alice's past records (0 to $t$) and future records ($t$ to $T$) | High-accuracy state estimation through post-hoc analysis | Bob's measurement data are not used in the estimation calculation |
| Reference state (true state) | Past records of both Alice and Bob (0 to $t$) | Reference for evaluating estimation accuracy | Derived from a precisely calibrated experimental model |
The smoothing calculation uses none of Bob's observation records. Alice uses only her own measurement data from before and after time $t$ and estimates the past quantum state inside the cavity based on the theoretical model. Bob's measurement data, by contrast, were used to obtain the reference state against which Alice's estimates were evaluated.
To calculate the reference state, all measurement records acquired by Alice and Bob up to time $t$ are used. These were fed into a physical model in which optical losses, detection efficiency and other parameters had been precisely measured and calibrated, yielding the state inside the cavity at time $t$. Because this reflects information that Alice alone cannot obtain, it allows a quantitative evaluation of how much the smoothing estimate improved.
However, this "true state" is not an absolute reference obtained by directly observing the inside of the cavity. It is a state derived on the premise of Alice's and Bob's measurement records and a calibrated experimental model. Its reliability depends on how accurately the optical constants, losses, detection efficiency and other properties of the experimental system are known.
Estimation errors in purity and squeezing shrink
In the experiment, the team tracked the time evolution of the quantum state inside the optical cavity while varying the measurement phase angles ( and ) of Alice's and Bob's homodyne detectors. For the electric-field components of light, which change on nanosecond (one-billionth of a second) timescales, they calculated estimated trajectories using both conventional filtering and smoothing that also uses later observation records. They then compared both with the trajectory of the reference true state to evaluate estimation accuracy.
The results confirmed that the mean-value trajectory obtained by smoothing was closer to the true state than the trajectory from filtering. For the covariance, which represents the uncertainty of the state, smoothing also kept the difference from the true state smaller. This experimentally shows that using later observation records improves the accuracy of estimates of past quantum states.
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"title": "量子状態推定における真の状態とのギャップ削減率",
"unit": "%",
"categories": ["状態の純度（Purity）", "スクイージングレベル（Squeezing）"],
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"caption": "それぞれ改善が最大となるよう検出条件を個別に最適化した測定点での値。誤差範囲は純度が±1.6%、スクイージングが±2.6%",
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}Two metrics were used to quantify estimation accuracy: "purity" and "squeezing level." Purity indicates how close a quantum state is to a pure state: it is 1 for a perfectly pure state and falls below 1 when noise or other effects make the state mixed. In filtering, some unobservable information remains, so the estimated state contains uncertainty and can be represented as a mixed state with lower purity. The squeezing level, meanwhile, indicates how much the quantum fluctuation in a particular quadrature component of the light is compressed compared with the vacuum state.
Varying the measurement conditions of Alice and Bob, the research group compared filtering and smoothing estimates for both metrics. Choosing the conditions that maximized the improvement for each metric, the gap from the true state shrank as follows.
- Purity improvement: Smoothing reduced the purity gap between the filtered estimate and the true state by 10.3 ± 1.6%.
- Squeezing improvement: The gap from the true state in squeezing level was likewise reduced by 7.6 ± 2.6%.
These figures do not mean that the actual purity of the light rose by 10.3% or that the squeezing strengthened by 7.6%. The physical state inside the optical cavity is not changed by post-hoc analysis; the figures indicate that the difference between the observer's estimated state and the true state became smaller. In addition, the absolute value of the gap used as the comparison baseline cannot be confirmed from the published materials.
The improvement rates of 10.3% and 7.6% are also not values obtained uniformly under all measurement conditions. They were obtained under conditions in which the detection angles of Alice and Bob were varied and individually optimized to maximize the improvement for purity and squeezing respectively. Therefore, the same rate of improvement is not necessarily obtained for quantum systems under different measurement conditions.
Linear Gaussian quantum systems and dependence on the model
The reason quantum state smoothing could be calculated analytically in this experiment is that the target was a "linear Gaussian quantum system." In 2019, Dr. Kiarn T. Laverick, Associate Professor Areeya Chantasri, Professor Wiseman and colleagues at the National University of Singapore and elsewhere derived a method for analytically obtaining smoothed states in quantum systems whose time evolution and measurement can be described by linear models and whose probability distribution remains Gaussian (normal) in form (DOI: 10.1103/PhysRevLett.122.190402).
The combination of a squeezed state inside an optical cavity and a homodyne detector is a typical example of a linear Gaussian quantum system. A Gaussian state can be fully described by just a finite number of parameters: a mean vector and a covariance matrix. This greatly simplifies calculation compared with estimating general quantum states, allowing smoothing with past and future observation records to be performed efficiently. The present demonstration exploits this mathematical property.
On the other hand, application to non-Gaussian states, which cannot be described by Gaussian distributions, has not been verified in this experiment. "Magic states," which are important in quantum computation, and Schrödinger's cat states are representative non-Gaussian quantum states. To handle such states, the calculation method used here, which describes a Gaussian state by its mean and covariance alone, cannot be applied as is. Estimation theories and computational methods aimed at more general quantum states are needed.
Precise modeling of the experimental system is also an important condition underpinning estimation accuracy. In this setup, how information unavailable from Alice's observation path is measured on Bob's side is understood and reflected in a physical model that includes optical losses and detection efficiency. In more complex quantum systems, however, the paths by which information flows out to the environment may be unknown, or the noise characteristics of measuring instruments may not be accurately known. It remains unclear whether the improvement in estimation accuracy from smoothing would be maintained in the presence of such model imperfections or calibration errors.
The result is therefore best regarded as the first experimental verification of the effectiveness of quantum state smoothing using a precisely controlled and calibrated optical cavity. How far results that hold in linear Gaussian quantum systems can be extended to non-Gaussian states or to quantum systems with complex interactions with the environment is left to future research.
Applications to post-hoc analysis such as measurement-based quantum computing
Because quantum state smoothing is a technique for analyzing records after measurement has finished, its main applications are post-hoc state estimation and device evaluation. In real-time control, future observation records that have not yet been obtained cannot be used, so smoothing cannot be incorporated as is into immediate feedback processing. This demonstration does not directly improve the physical sensitivity of quantum sensors or the computational speed of quantum computers.
One field where applications are anticipated is the analysis of states in measurement-based quantum computers. In this approach, a cluster state in which many qubits are entangled is prepared in advance, and computation proceeds by measuring the qubits one after another. Because subsequent measurement conditions and logical operations change depending on the measurement results, it is important to properly understand the quantum state midway through the computation. If the series of measurement records can be analyzed after the computation ends, it could provide a clue for evaluating whether the quantum state evolved as designed at a particular stage.
By using not only past measurement records but also those obtained afterward, it may be possible to reconstruct the quantum state and purity during a computation more precisely than before. Furthermore, if the way decoherence changed the quantum state can be analyzed after the fact, it is thought that this could also help evaluate error-correction protocols and investigate the causes of errors arising during computation. However, these applications were not directly verified in the present optical cavity experiment, and demonstrating their effectiveness in actual measurement-based quantum computation will require extensions of theory and experiment suited to the relevant quantum states and measurement processes.
The essence of quantum state smoothing is to use observation records obtained at later times to improve estimates of past states in accordance with the principles of quantum mechanics. This study quantitatively demonstrated that effect in an optical experiment on a linear Gaussian quantum system. Going forward, what will matter is extending the method to non-Gaussian states, testing its robustness against model imperfections and measurement errors, and determining whether it can be used to evaluate real quantum devices and analyze computational processes.