Squeezing Information from Light: Reaching Precision Limits in Quantum Measurement

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Author: Denis Avetisyan

A new theoretical framework demonstrates how leveraging squeezed states during dispersive readout can unlock optimal precision for estimating resonator frequency in quantum systems.

The dispersive readout of a resonator-enhanced by both vacuum squeezing and antisqueezing via a phase-sensitive amplifier-demonstrates that increasing nonlinearity beyond a certain point yields diminishing returns in cumulative Fisher information, while squeezing depth exponentially increases information acquisition until it converges with the quantum Fisher information limit, as evidenced by parameter regimes of $\beta^{2}=10\Omega$ and $\gamma=0.4\Omega$.
The dispersive readout of a resonator-enhanced by both vacuum squeezing and antisqueezing via a phase-sensitive amplifier-demonstrates that increasing nonlinearity beyond a certain point yields diminishing returns in cumulative Fisher information, while squeezing depth exponentially increases information acquisition until it converges with the quantum Fisher information limit, as evidenced by parameter regimes of $\beta^{2}=10\Omega$ and $\gamma=0.4\Omega$.

Combining full-counting statistics and quantum trajectories reveals the potential for quantum-optimal parameter estimation using squeezed states in dispersive readout.

Achieving quantum-limited precision in continuous measurement remains a significant challenge despite advances in quantum metrology. This is addressed in ‘Full-counting statistics and quantum information of dispersive readout with a squeezed environment’, where we develop a framework to analyze dispersive readout enhanced by squeezed states. Our analysis, utilizing full-counting statistics and a non-Hermitian mean-field approach, demonstrates that this setup can approach quantum-optimal precision for resonator frequency estimation, exhibiting robustness against system nonlinearities. Could this streamlined theoretical approach facilitate the wider adoption of squeezed-state enhanced dispersive readout in practical quantum technologies?

The Delicate Balance of Observation

The pursuit of enhanced quantum sensing and control fundamentally hinges on a delicate balance: minimizing disturbance during measurement. Any attempt to observe a quantum system inevitably introduces backaction – a perturbation that alters the very state being measured. Simultaneously, inherent noise, arising from both the environment and the measurement apparatus itself, obscures the signal and limits precision. Therefore, advanced techniques focus on reducing both backaction and noise to approach the fundamental limits of quantum metrology, allowing for increasingly accurate determination of system properties and more reliable control over quantum dynamics. This necessitates innovative approaches to measurement, often involving weak or non-demolition schemes designed to extract information with minimal disruption, and sophisticated noise filtering strategies to isolate the desired signal from unwanted fluctuations.

Dispersive readout, a cornerstone of quantum measurement, frequently employs the Input-Output Formalism to infer a quantum system’s state. However, this approach carries inherent limitations when confronted with complex quantum dynamics. The formalism often relies on approximations – such as treating the probe field as weakly coupled and assuming a linear response – which begin to falter as systems exhibit stronger interactions or non-linear behavior. These simplifications can obscure crucial details of the quantum evolution, leading to inaccurate reconstructions of the system’s state and hindering the ability to fully characterize phenomena like entanglement or many-body interactions. Consequently, while effective for simple systems, standard dispersive readout may fail to capture the richness of complex quantum processes, necessitating more sophisticated measurement strategies and theoretical frameworks to overcome these limitations and unlock the full potential of quantum sensing and control.

The precision of quantum measurements is fundamentally constrained by the simplifications inherent in modeling the interaction between the measuring device – the probe – and the quantum system it aims to observe. Standard approaches often rely on approximations, such as treating the probe-system interaction as weak or assuming a specific form for the system’s dynamics, to make calculations tractable. However, these approximations can obscure crucial details of the quantum evolution, particularly in scenarios involving strong coupling or complex system behavior. Consequently, the resulting measurement models may fail to accurately capture the full extent of quantum backaction-the disturbance introduced by the measurement itself-and introduce systematic errors that limit the achievable precision. Addressing these limitations requires developing more sophisticated theoretical frameworks and measurement techniques that can faithfully represent the intricacies of the probe-system interaction and minimize the impact of simplifying assumptions.

Fisher information scales with the Kerr nonlinearity parameter U2U_2, consistent with the overall parameter settings established in Figure 1.
Fisher information scales with the Kerr nonlinearity parameter U2U_2, consistent with the overall parameter settings established in Figure 1.

Simulating Quantum Reality with Trajectories

The Quantum Trajectory Formalism provides a method for simulating the time evolution of a quantum system subject to continuous measurement. This formalism represents the system’s wavefunction as an ensemble of stochastic trajectories, each corresponding to a possible measurement outcome. Each trajectory evolves according to a non-unitary Schrödinger equation driven by the measurement process, effectively ‘collapsing’ the wavefunction incrementally with each infinitesimally small measurement. The probability of each trajectory is determined by the associated measurement outcome, and the overall system dynamics are obtained by averaging over all trajectories, weighted by their respective probabilities. This approach allows for the modeling of wavefunction dynamics under realistic, continuous measurement scenarios, unlike standard solutions to the Schrödinger equation which assume no measurement or instantaneous, discrete measurements.

The Quantum Trajectory Formalism models the readout process as a stochastic process, acknowledging that measurement inherently introduces randomness into the system’s evolution. This is crucial because quantum fluctuations, arising from the Heisenberg uncertainty principle, are not simply noise but fundamental aspects of the quantum state. The formalism represents the system’s wavefunction as an ensemble of trajectories, each evolving according to a stochastic Schrödinger equation. The probability of each trajectory is determined by the measurement record, effectively weighting trajectories based on their consistency with observed data. This allows for the calculation of expectation values and other observables by averaging over these trajectories, thereby incorporating the impact of quantum fluctuations and the stochasticity of the measurement process into the system’s dynamics. The resulting dynamics are non-unitary, reflecting the information loss associated with measurement and the influence of the measurement apparatus on the quantum system.

Extending the Quantum Trajectory Formalism with Non-Hermitian Mean-Field Theory enables the analysis of system dynamics over extended timescales, particularly relevant for continuous measurement scenarios. This approach introduces an effective Hamiltonian that incorporates the influence of the measurement apparatus, represented by a non-Hermitian term. By solving the time-dependent Schrödinger equation with this effective Hamiltonian, the evolution of the quantum state can be predicted. The non-Hermitian term accounts for the loss of quantum coherence due to the continuous extraction of information, allowing for the calculation of the system’s steady-state properties and the prediction of measurement outcomes, including the average current and noise characteristics, as a function of the measurement strength and timescale. This methodology provides a computationally tractable means of investigating the impact of continuous observation on quantum systems, overcoming limitations inherent in traditional approaches that struggle with long-time dynamics and decoherence effects.

Harnessing Quantum Fluctuations for Precision

A squeezed environment reduces quantum noise by redistributing the uncertainty between quadrature phases of an electromagnetic field. Standard quantum mechanics dictates a lower bound on the product of uncertainties in conjugate variables; squeezing minimizes the uncertainty in one quadrature phase at the expense of increased uncertainty in the other. This noise reduction directly translates to enhanced measurement sensitivity for parameters coupled to the squeezed field, allowing for the detection of weaker signals and more precise estimations of physical quantities. Specifically, the signal-to-noise ratio for measurements is improved, scaling with the degree of squeezing achieved; a higher degree of squeezing yields a proportionally greater enhancement in precision, down to and potentially below the standard quantum limit, which is defined by $ \frac{1}{\sqrt{N}} $, where N is the number of particles.

The Bogoliubov transformation is employed to mathematically describe the effect of squeezing on the quadrature amplitudes of the electromagnetic field within the optical resonator. This transformation effectively rotates the zero-point fluctuations, reducing the noise in one quadrature at the expense of increased noise in the conjugate quadrature. Specifically, the transformation maps the original annihilation and creation operators, $a$ and $a^\dagger$, to squeezed operators, $a_{sq}$ and $a_{sq}^\dagger$, allowing for a precise quantification of the noise reduction and subsequent improvement in measurement sensitivity. By tailoring the parameters of the Bogoliubov transformation, the measurement process can be optimized to maximize the signal-to-noise ratio for the parameter being estimated, leading to enhanced precision beyond the standard quantum limit.

The generation of a squeezed state, essential for enhancing measurement precision, is fundamentally dependent on the preservation of Time-Reversal Symmetry (TRS) within the experimental apparatus. TRS, mathematically defined by the invariance of physical laws under the transformation $t \rightarrow -t$, dictates the allowable interactions and energy flow within the system. Non-reciprocal elements, such as circulators or certain types of Josephson junctions, break TRS and can introduce excess noise or prevent the formation of the desired squeezed state. Specifically, the Bogoliubov transformation, used to describe squeezing, relies on the Hermitian nature of the Hamiltonian, which is guaranteed only when TRS is maintained. Therefore, careful design and calibration are required to ensure all components operate within the constraints of TRS to achieve a properly engineered squeezed state and maximize the benefits of reduced quantum noise.

The first four cumulants exhibit a dependence on detuning that varies with squeezing strength, consistent with the parameters used in the previous figure.
The first four cumulants exhibit a dependence on detuning that varies with squeezing strength, consistent with the parameters used in the previous figure.

The Limits of Knowability and the Pursuit of Precision

Fisher Information serves as a fundamental metric for evaluating the sensitivity of a measurement to changes in a parameter being estimated. This quantity directly relates to the precision with which that parameter can be determined; higher Fisher Information signifies a greater ability to distinguish between similar parameter values, and therefore, improved estimation accuracy. Researchers utilize Fisher Information not only to assess the performance of existing measurement schemes, but also to guide the development of optimal strategies. By maximizing Fisher Information, one can, in principle, design measurements that extract the most information possible from a given signal, thereby achieving the theoretical limits of precision. This approach allows for a systematic comparison of different measurement techniques and provides a pathway towards enhancing the accuracy of parameter estimation in diverse scientific applications, ranging from spectroscopy to gravitational wave detection. The calculation involves analyzing the probability distribution of the measurement outcomes and quantifying how much it changes with respect to the parameter of interest, often expressed as $F = E[\left(\frac{\partial}{\partial \theta} \log P(x|\theta)\right)^2]$.

The utility of Fisher Information extends beyond simply quantifying measurement sensitivity; its connection to statistical moments, specifically cumulants, provides a powerful means to characterize the underlying probability distribution of estimated parameters. Cumulants, which describe the shape and spread of a distribution – including skewness and kurtosis – are directly related to derivatives of the log-likelihood function, and thus to the Fisher Information. By analyzing these relationships, researchers can gain insights into the quality of parameter estimates, identifying potential biases or inaccuracies. A larger Fisher Information generally corresponds to a narrower distribution – indicating higher precision – while the cumulants reveal details about the distribution’s symmetry and peakedness. This interplay allows for a comprehensive assessment of estimation quality, moving beyond simple variance calculations to a richer understanding of the entire probability landscape and enabling the development of more robust and reliable estimation strategies.

The study reveals a significant enhancement in measurement sensitivity directly linked to the squeezing parameter, denoted as ‘r’. Specifically, the Fisher Information – a key metric for quantifying estimation precision – scales exponentially with ‘r’, achieving a factor of $e^{2r}$. This implies that even modest increases in squeezing can dramatically improve the ability to discern small changes in the measured parameter. Further analysis at zero detuning reveals a precise relationship for the Fisher Information: $64Ω²γ⁻³e²r$, where Ω represents the Rabi frequency, γ the decay rate, and ‘e’ is a constant related to the elementary charge. This detailed scaling demonstrates how manipulating these parameters, in conjunction with squeezing, can optimize measurement precision and approach fundamental limits.

The fundamental limit of precision in parameter estimation is defined by the Quantum Fisher Information (QFI), and this analysis establishes a pathway towards achieving that limit through optimized measurement strategies. By comparing the Classical Fisher Information (CFI) with the QFI, a quantifiable metric – termed quantum efficiency – reveals how closely a given measurement approach aligns with quantum optimality. Results demonstrate that increasing the squeezing parameter, $r$, drives the quantum efficiency towards unity, indicating a convergence towards the ultimate precision bounds dictated by quantum mechanics. This suggests that with sufficient squeezing, the developed measurement scheme can effectively harness quantum resources to minimize estimation uncertainty, offering a robust method for approaching the theoretical limits of precision in parameter estimation.

The pursuit of quantum-optimal precision, as detailed within this study of dispersive readout, echoes a fundamental principle of all systems: their eventual degradation. While the research highlights methods to delay this decay – leveraging squeezed states to minimize estimation error and approach the theoretical limit – it implicitly acknowledges the transient nature of stability. As Albert Einstein observed, “The only thing that you must learn in life is that you must learn to learn.” This resonates deeply with the work; the continuous refinement of measurement techniques, striving for ever-greater accuracy, is itself an acknowledgement that even optimized systems are subject to the relentless march of time and require constant adaptation. The theoretical framework presented isn’t about achieving permanent precision, but about maximizing it within the constraints of inherent latency and eventual decay.

What Lies Ahead?

The pursuit of quantum optimality, as demonstrated through the lens of dispersive readout with squeezed states, reveals not an endpoint, but a shifting horizon. Every commit in this annals of research, every version of theoretical refinement, records a chapter in understanding the inevitable decay inherent in any measurement process. The framework presented – a confluence of quantum trajectories and full-counting statistics – offers a powerful diagnostic, yet it does not suspend the second law. Future iterations must address the practical tax levied by imperfect state preparation and the decoherence that erodes the squeezed environment’s advantage.

A pressing question lingers: how robust is this enhanced precision against realistic noise spectra? The theoretical scaffolding currently prioritizes a carefully sculpted environment. Delaying fixes for these imperfections is, in effect, a tax on ambition. Further study should explore the interplay between squeezing, noise, and the limitations imposed by non-Hermitian mean-field approximations.

Ultimately, the true measure of progress lies not in approaching an idealized limit, but in gracefully navigating the inevitable entropic drift. The path forward demands a more holistic accounting – not merely of signal enhancement, but of the total cost – in energy, complexity, and resilience – of extracting information from a quantum system.

Original article: https://arxiv.org/pdf/2512.02531.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2025-12-03 17:05