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arXiv:1903.02563 · Nature Communications

Quantum advantage in postselected metrology

David R. M. Arvidsson-Shukur, Nicole Yunger Halpern, Hugo Vincent Lepage, Aleksander Lasek, Crispin H. W. Barnes, Seth Lloyd

Published Jul 29, 2020• 141 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1038/s41467-020-17559-w

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Quantum advantage in postselected metrology

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Abstract: 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.

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Comment on “Saturating quantum advantages in postselected metrology with the positive Kirkwood-Dirac distribution”

David R. M. Arvidsson-Shukur, C. H. W. Barnes, Hugo Vincent Lepage +4 more

In a recent paper [Das, Modak, and Bera, Phys. Rev. A 107, 042413 (2023)], Das et al. claim that Kirkwood-Dirac negativity is not required to achieve anomalous values of the quantum Fisher information encoded in postselected (filtered) particles. We argue that this claim is wrong. Further, we show that the quantum Fisher information in optimal postselected metrology is directly proportional to the total Kirkwood-Dirac nonclassicality.

Physical Review A•Jun 21, 2024•3 citations
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.
Abstract: 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.
Weak amplification and other postselection-based metrological protocols can enhance precision while estimating small parameters, outperforming postselection-free protocols. In general, these enhancements are largely constrained because the protocols yielding higher precision are rarely obtained due to a lower probability of successful postselection. It is shown that this precision can further be improved with the help of quantum resources like entanglement and negativity in the quasiprobability distribution. However, these quantum advantages in attaining considerable success probability with large precision are bounded irrespective of any accessible quantum resources. The advantage is being considered only within the scope of postselected metrology. Here we derive a bound of these advantages in postselected metrology, establishing a connection with weak value optimization where the latter can be understood in terms of the geometric phase. We introduce a scheme that saturates the bound, yielding anomalously large precision. Moreover, we prove that these advantages can be achieved with positive quasiprobability distribution. We also provide an optimal metrological scheme using a three-level nondegenerate quantum system.
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.
In conventional measurement, to reach the greatest accuracy of parameter estimation, all samples must be measured since each independent sample contains the same quantum Fisher information. In postselected metrology, postselection can concentrate the quantum Fisher information of the initial samples into a tiny post-selected sub-ensemble. It has been proven that this quantum advantage can not be realized in any classically commuting theory. In this work, we present that the advantage of postselection in weak value amplification (WVA) can not be achieved without quantum coherence. The quantum coherence of the initial system is closely related to the preparation costs and measurement costs in parameter estimation. With the increase of initial quantum coherence, the joint values of preparation costs and measurement costs can be optimized to smaller. Moreover, we derive an analytical tradeoff relation between the preparation, measurement and the quantum coherence. We further experimentally test the tradeoff relation in a linear optical setup. The experimental and theoretical results are in good agreement and show that the quantum coherence plays a key role in bounding the resource costs in the postselected metrology process.
Negative Kirkwood--Dirac quasiprobabilities are credited with the quantum advantage of interaction-free measurement and postselected metrology. We test that claim on the task interaction-free measurement was designed for: learning about a sample letting it absorb as little light as possible. The Kirkwood--Dirac distribution of path and output gives the linear response to a weak sample: its real part responds to absorption, its imaginary part to a phase. For a pure single photon passing once through any interferometer, the Fisher information about transmittance per absorbed photon is the direct-transmission value minus a sum of squared imaginary parts: no interferometer or readout beats direct transmission. Negativity plays no role beyond concentrating information in weakly lit outputs. The imaginary parts of the Kirkwood-Dirac distribution depend on the output basis here, not on the sample, so a basis exists in which every term is real, which is optimal. A single photon gains information per absorbed photon only through a phase response, which for an optical line needs a thick or inhomogeneously broadened sample, or through many passes when absorptions go uncounted, as shown by a critically coupled resonator. For detecting a weak absorber, negativity gives a single-shot advantage that does not accumulate over multiple trials; advantage instead accumulates when there are outputs which are dark without the object, a case which needs no negativity.