Improving the Precision of Weak Measurements by Postselection Measurement

Agnostic Phase Estimation

The goal of quantum metrology is to improve measurements' sensitivities by harnessing quantum resources. Metrologists often aim to maximize the quantum Fisher information, which bounds the measurement setup's sensitivity. In studies of fundamental limits on metrology, a paradigmatic setup features a qubit (spin-half system) subject to an unknown rotation. One obtains the maximal quantum Fisher information about the rotation if the spin begins in a state that maximizes the variance of the rotation-inducing operator. If the rotation axis is unknown, however, no optimal single-qubit sensor can be prepared. Inspired by simulations of closed timelike curves, we circumvent this limitation. We obtain the maximum quantum Fisher information about a rotation angle, regardless of the unknown rotation axis. To achieve this result, we initially entangle the probe qubit with an ancilla qubit. Then, we measure the pair in an entangled basis, obtaining more information about the rotation angle than any single-qubit sensor can achieve. We demonstrate this metrological advantage using a two-qubit superconducting quantum processor. Our measurement approach achieves a quantum advantage, outperforming every entanglement-free strategy.

Theory of Compression Channels for Postselected Quantum Metrology

The postselected quantum metrological scheme is especially advantageous when the final measurements are either very noisy or expensive in practical experiments. In this Letter, we put forward a general theory on the compression channels in postselected quantum metrology. We define the basic notions characterizing the compression quality and illuminate the underlying structure of lossless compression channels. Previous experiments on postselected optical phase estimation and weak-value amplification are shown to be particular cases of this general theory. Furthermore, for two categories of bipartite systems, we show that the compression loss can be made arbitrarily small even when the compression channel acts only on one subsystem. These findings can be employed to distribute quantum measurements so that the measurement noise and cost are dramatically reduced.

Quantum-coherence-free precision metrology by means of difference-signal amplification

The novel weak-value-amplification (WVA) scheme of precision metrology is deeply rooted in the quantum nature of destructive interference between the pre- and post-selection states. And, an alternative version, termed as joint WVA (JWVA), which employs the difference-signal from the post-selection accepted and rejected results, has been found possible to achieve even better sensitivity (two orders of magnitude higher) under some technical limitations (e.g. misalignment errors). In this work, after erasing the quantum coherence, we analyze the difference-signal amplification (DSA) technique, which serves as a classical counterpart of the JWVA, and show that similar amplification effect can be achieved. We obtain a simple expression for the amplified signal, carry out characterization of precision, and point out the optimal working regime. We also discuss how to implement the post-selection of a classical mixed state. The proposed classical DSA technique holds similar technical advantages of the JWVA and may find interesting applications in practice.

Negative Quasiprobabilities Enhance Phase Estimation in Quantum-Optics Experiment

Operator noncommutation, a hallmark of quantum theory, limits measurement precision, according to uncertainty principles. Wielded correctly, though, noncommutation can boost precision. A recent foundational result relates a metrological advantage with negative quasiprobabilities-quantum extensions of probabilities-engendered by noncommuting operators. We crystallize the relationship in an equation that we prove theoretically and observe experimentally. Our proof-of-principle optical experiment features a filtering technique that we term partially postselected amplification (PPA). Using PPA, we measure a wave plate's birefringent phase. PPA amplifies, by over two orders of magnitude, the information obtained about the phase per detected photon. In principle, PPA can boost the information obtained from the average filtered photon by an arbitrarily large factor. The filter's amplification of systematic errors, we find, bounds the theoretically unlimited advantage in practice. PPA can facilitate any phase measurement and mitigates challenges that scale with trial number, such as proportional noise and detector saturation. By quantifying PPA's metrological advantage with quasiprobabilities, we reveal deep connections between quantum foundations and precision measurement.

Heisenberg-Limited Metrology via Weak-Value Amplification without Using Entangled Resources

Weak-value amplification (WVA) provides a way for amplified detection of a tiny physical signal at the expense of a lower detection probability. Despite this trade-off, due to its robustness against certain types of noise, WVA has advantages over conventional measurements in precision metrology. Moreover, it has been shown that WVA-based metrology can reach the Heisenberg limit using entangled resources, but preparing macroscopic entangled resources remains challenging. Here, we demonstrate a novel WVA scheme based on iterative interactions, achieving the Heisenberg-limited precision scaling without resorting to entanglement. This indicates that the perceived advantages of the entanglement-assisted WVA are in fact due to iterative interactions between each particle of an entangled system and a meter, rather than coming from the entanglement itself. Our work opens a practical pathway for achieving the Heisenberg-limited WVA without using fragile and experimentally demanding entangled resources.

Detection of magneto-optical Kerr signals via weak measurement with frequency pointer

Detection of the magneto-optical Kerr effect with high precision is of great significance but has challenges in the field of magnetic physics and spintronic devices. Kerr rotation angle and Kerr ellipticity always coexist and are difficult to distinguish, which jointly determines the light intensity received by the detector and limits the improvement of measurement precision. In this Letter, a nonlinear weak measurement scheme for magneto-optical Kerr signals with a frequency pointer is proposed. The Kerr rotation and Kerr ellipticity can be separately detected by constructing different pre-selections and choosing the appropriate coupling strength. Moreover, two signals obtained through the weak measurement scheme have higher precision and signal-to-noise ratio compared with the standard polarimetry scheme. Our method may have important applications in the field of magneto-optic parameters measurement or magnetic sensors.

Beating Standard Quantum Limit with Weak Measurement

Weak measurements have been under intensive investigation in both experiment and theory. Numerous experiments have indicated that the amplified meter shift is produced by the post-selection, yielding an improved precision compared to conventional methods. However, this amplification effect comes at the cost of a reduced rate of acquiring data, which leads to an increasing uncertainty to determine the level of meter shift. From this point of view, a number of theoretical works have suggested that weak measurements cannot improve the precision, or even damage the metrology information due to the post-selection. In this review, we give a comprehensive analysis of the weak measurements to justify their positive effect on prompting measurement precision. As a further step, we introduce two modified weak measurement protocols to boost the precision beyond the standard quantum limit. Compared to previous works beating the standard quantum limit, these protocols are free of using entangled or squeezed states. The achieved precision outperforms that of the conventional method by two orders of magnitude and attains a practical Heisenberg scaling up to n=106 photons.

Approaching Quantum-Limited Metrology with Imperfect Detectors by Using Weak-Value Amplification

Weak-value amplification (WVA) is a metrological protocol that amplifies ultrasmall physical effects. However, the amplified outcomes necessarily occur with highly suppressed probabilities, leading to the extensive debate on whether the overall measurement precision is improved in comparison to that of conventional measurement (CM). Here, we experimentally demonstrate the unambiguous advantages of WVA that overcome practical limitations including noise and saturation of photodetection and maintain a shot-noise-scaling precision for a large range of input light intensity well beyond the dynamic range of the photodetector. The precision achieved by WVA is 6 times higher than that of CM in our setup. Our results clear the way for the widespread use of WVA in applications involving the measurement of small signals including precision metrology and commercial sensors.

Quantum advantage in postselected metrology

In every parameter-estimation experiment, the final measurement or the postprocessing incurs a cost. Postselection can improve the rate of Fisher information (the average information learned about an unknown parameter from a trial) to cost. We show that this improvement stems from the negativity of a particular quasiprobability distribution, a quantum extension of a probability distribution. In a classical theory, in which all observables commute, our quasiprobability distribution is real and nonnegative. In a quantum-mechanically noncommuting theory, nonclassicality manifests in negative or nonreal quasiprobabilities. Negative quasiprobabilities enable postselected experiments to outperform optimal postselection-free experiments: postselected quantum experiments can yield anomalously large information-cost rates. This advantage, we prove, is unrealizable in any classically commuting theory. Finally, we construct a preparation-and-postselection procedure that yields an arbitrarily large Fisher information. Our results establish the nonclassicality of a metrological advantage, leveraging our quasiprobability distribution as a mathematical tool.

Measurement of hysteresis loop based on weak measurement

In this Letter, we propose a technique for hysteresis loop measurement based on weak measurement. By using the photonic spin Hall effect (PSHE) as a probe and combining the quantum weak measurement, the technique's noise can be suppressed greatly. A theoretical model to describe the numerical relation between the amplified shift and Kerr rotation angle is established. Through detecting the amplified shift of the PSHE based on weak measurement, we experimentally measure the hysteresis loops of Ni-Fe alloy film, iron-phthalocyanine (FePc) monolayer film, and Co/FePc double-layer film. The results show that the precision can reach about $ \sim {10^{ - 6}} \;{\rm rad} $∼10^-6rad under ordinary experimental conditions, which may have an important application prospect in magneto-optic parameters measurement.

Optimization of a quantum weak measurement system with digital filtering technology

In this paper, we propose a post-Gaussian filtering theory for weak measurement in the frequency domain, and propose a highly deformed digital filtering technique that can be used to optimize sensors based on weak-frequency measurement techniques. We completed the experimental verification based on the weak measurement total internal reflection sensor. The experimental results show that digital filtering technology can optimize the system in the working range, sensitivity, and resolution of the frequency domain weak measurement system, so that it can reach 0.210 rad, 3210.9 nm/RIU, and 7.12×10^-7  RIU, respectively.

Achieving Heisenberg-Scaling Precision with Projective Measurement on Single Photons

It has been suggested that both quantum superpositions and nonlinear interactions are important resources for quantum metrology. However, to date the different roles that these two resources play in the precision enhancement are not well understood. Here, we experimentally demonstrate a Heisenberg-scaling metrology to measure the parameter governing the nonlinear coupling between two different optical modes. The intense mode with n (more than 10^{6} in our work) photons manifests its effect through the nonlinear interaction strength which is proportional to its average photon number. The superposition state of the weak mode, which contains only a single photon, is responsible for both the linear Hamiltonian and the scaling of the measurement precision. By properly preparing the initial state of single photon and making projective photon-counting measurements, the extracted classical Fisher information (FI) can saturate the quantum FI embedded in the combined state after coupling, which is ∼n^{2} and leads to a practical precision ≃1.2/n. Free from the utilization of entanglement, our work paves a way to realize Heisenberg-scaling precision when only a linear Hamiltonian is involved.

Optimization of a quantum weak measurement system with its working areas

Phase-sensitive weak measurement systems have been receiving an increasing amount of attention. In this paper, we introduce a series of weak measurement working areas. By adjusting the pre-selection and post-selection states and the total phase difference between vertically polarized light and horizontally polarized light, the measurement of the weak value is amplified by several times in one system. Its applicability is verified in a label-free total internal reflection system. The original sensitivity and resolution are improved at different working areas, reaching 1.85 um/refractive index unit (RIU) and 6.808 × 10^-7 RIU, respectively.

Heisenberg-scaling measurement of the single-photon Kerr non-linearity using mixed states

Improving the precision of measurements is a significant scientific challenge. Previous works suggest that in a photon-coupling scenario the quantum fisher information shows a quantum-enhanced scaling of N², which in theory allows a better-than-classical scaling in practical measurements. In this work, utilizing mixed states with a large uncertainty and a post-selection of an additional pure system, we present a scheme to extract this amount of quantum fisher information and experimentally attain a practical Heisenberg scaling. We performed a measurement of a single-photon's Kerr non-linearity with a Heisenberg scaling, where an ultra-small Kerr phase of ≃6 × 10^-8 rad was observed with a precision of ≃3.6 × 10^-10 rad. From the use of mixed states, the upper bound of quantum fisher information is improved to 2N². Moreover, by using an imaginary weak-value the scheme is robust to noise originating from the self-phase modulation.

Weak value controversy

Recent controversy regarding the meaning and usefulness of weak values is reviewed. It is argued that in spite of recent statistical arguments by Ferrie and Combes, experiments with anomalous weak values provide useful amplification techniques for precision measurements of small effects in many realistic situations. The statistical nature of weak values is questioned. Although measuring weak values requires an ensemble, it is argued that the weak value, similarly to an eigenvalue, is a property of a single pre- and post-selected quantum system.This article is part of the themed issue 'Second quantum revolution: foundational questions'.

Weak Value Amplification Can Outperform Conventional Measurement in the Presence of Detector Saturation

Weak value amplification (WVA) is a technique by which one can magnify the apparent strength of a measurement signal. Some have claimed that WVA can outperform more conventional measurement schemes in parameter estimation. Nonetheless, a significant body of theoretical work has challenged this perspective, suggesting WVA to be fundamentally suboptimal. Optimal measurements may not be practical, however. Two practical considerations that have been conjectured to afford a benefit to WVA over conventional measurement are certain types of noise and detector saturation. Here, we report a theoretical study of the role of saturation and pixel noise in WVA-based measurement, in which we carry out a Bayesian analysis of the Fisher information available using a saturable, pixelated, digitized, and/or noisy detector. We draw two conclusions: first, that saturation alone does not confer an advantage to the WVA approach over conventional measurement, and second, that WVA can outperform conventional measurement when saturation is combined with intrinsic pixel noise and/or digitization.

Experimental Demonstration of Higher Precision Weak-Value-Based Metrology Using Power Recycling

The weak-value-based metrology is very promising and has attracted a lot of attention in recent years because of its remarkable ability in signal amplification. However, it is suggested that the upper limit of the precision of this metrology cannot exceed that of classical metrology because of the low sample size caused by the probe loss during postselection. Nevertheless, a recent proposal shows that this probe loss can be reduced by the power-recycling technique, and thus enhance the precision of weak-value-based metrology. Here we experimentally realize the power-recycled interferometric weak-value-based beam-deflection measurement and obtain the amplitude of the detected signal and white noise by discrete Fourier transform. Our results show that the detected signal can be strengthened by power recycling, and the power-recycled weak-value-based signal-to-noise ratio can surpass the upper limit of the classical scheme, corresponding to the shot-noise limit. This work sheds light on higher precision metrology and explores the real advantage of the weak-value-based metrology over classical metrology.

Optical weak measurement system with common path implementation for label-free biomolecule sensing

A reflection-type phase-sensitive weak measurement for biosensing and chemical label-free sensing is presented. The phase difference between p and s polarizations in total internal reflection caused by biomolecular recognition is measured by weak value amplification. The system with p and s polarizations in a common path is stable and robust. The sensing process occurring on the silicon dioxide surface is achieved with a resolution of 3.6×10^-6 refractive index units. The applicability is demonstrated by real-time monitoring biomolecular interaction of IgG and protein A.

Metrology with Unknown Detectors

The best possible precision is one of the key figures in metrology, but this is established by the exact response of the detection apparatus, which is often unknown. There exist techniques for detector characterization that have been introduced in the context of quantum technologies but apply as well for ordinary classical coherence; these techniques, though, rely on intense data processing. Here, we show that one can make use of the simpler approach of data fitting patterns in order to obtain an estimate of the Cramér-Rao bound allowed by an unknown detector, and we present applications in polarimetry. Further, we show how this formalism provides a useful calculation tool in an estimation problem involving a continuous-variable quantum state, i.e., a quantum harmonic oscillator.