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Quantum Advantage in Postselected Metrology ✉ David R

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Quantum Advantage in Postselected Metrology ✉ David R ARTICLE https://doi.org/10.1038/s41467-020-17559-w OPEN Quantum advantage in postselected metrology ✉ David R. M. Arvidsson-Shukur 1,2,3 , Nicole Yunger Halpern 3,4,5, Hugo V. Lepage 1, Aleksander A. Lasek 1, Crispin H. W. Barnes1 & Seth Lloyd2,3 In every parameter-estimation experiment, the final measurement or the postprocessing incurs a cost. Postselection can improve the rate of Fisher information (the average infor- mation 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 1234567890():,; 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 quasi- probabilities. 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 proce- dure that yields an arbitrarily large Fisher information. Our results establish the non- classicality of a metrological advantage, leveraging our quasiprobability distribution as a mathematical tool. 1 Cavendish Laboratory, Department of Physics, University of Cambridge, Cambridge CB3 0HE, UK. 2 Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. 3 Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. 4 ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA. 5 Department of Physics, Harvard University, Cambridge, MA ✉ 02138, USA. email: [email protected] NATURE COMMUNICATIONS | (2020) 11:3775 | https://doi.org/10.1038/s41467-020-17559-w | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-17559-w ur ability to deliver new quantum-mechanical improve- 1–p ments to technologies relies on a better understanding of Γ O p ps the foundation of quantum theory: When is a phenom- F ( ) enon truly nonclassical? We take noncommutation as our notion Fig. 1 Classical experiment with postselection. A nonoptimal input device of nonclassicality and we quantify this nonclassicality with initializes a particle in one of two states, with probabilities p and 1 − p, negativity: Quantum states can be represented by quasiprobability respectively. The particle undergoes a transformation Γθ set by an unknown distributions, extensions of classical probability distributions. parameter θ. Only the part of the transformation that acts on particles in Whereas probabilities are real and nonnegative, quasiprobabilities the lower path depends on θ. If the final measurement is expensive, the can assume negative and nonreal values. Quasiprobabilities’ negativity stems from the impossibility of representing quantum particles in the upper path should be discarded: they possess no information about θ. states with joint probability distributions1–3. The distribution we use, an extension of the Kirkwood–Dirac distribution4–6, signals nonclassical noncommutation through the presence of negative or nonreal quasiprobabilities. an optimal input and final measurement can. We thus conclude One field advanced by quantum mechanics is metrology, which that experiments that generate anomalous Fisher-information concerns the statistical estimation of unknown physical para- values require noncommutativity. meters. Quantum metrology relies on quantum phenomena to improve estimations beyond classical bounds7. A famous example Results exploits entanglement8–10. Consider using N separable and dis- Postselected quantum Fisher information. As aforementioned, tinguishable probe states to evaluate identical systems in parallel. postselection can raise the Fisher information per final mea- The best estimator’s error will scale as N−1/2. If the probes are surement. Figure 1 outlines a classical experiment with such an entangled, the error scaling improves to N−1 11. As Bell’s theorem information enhancement. Below, we show how postselection rules out classical (local realist) explanations of entanglement, the affects the Fisher information in a quantum setting. improvement is genuinely quantum. Consider an experiment with outcomes i and associated θ A central quantity in parameter estimation is the Fisher probabilities pi( ), which depend on some unknown parameter information, IðθÞ. The Fisher information quantifies the θ. The Fisher information about θ is14 average information learned about an unknown parameter θ X X 12–14 θ 2 1 2 from an experiment . Ið Þ lower-bounds the variance of an IðθÞ¼ p ðθÞ½∂θln ðp ðθÞÞ ¼ ½∂θp ðθÞ : ð1Þ θ – i i p ðθÞ i unbiased estimator e via the Cramér Rao inequality: i i i θ ≥ = θ 15,16 Varð eÞ 1 Ið Þ . A common metrological task concerns ≫ optimally estimating a parameter that characterizes a physical Repeating the experiment N 1 times provides, on average, an fi amount NIðθÞ of information about θ. The estimator’s variance process. The experimental input and the nal measurement are θ ≥ = θ is bounded by Varð eÞ 1 ½NIð Þ. optimized to maximize the Fisher information and to minimize fi the estimator’s error. Below, we de ne and compare two types of metrological fi procedures. In both scenarios, we wish to estimate an unknown Classical parameter estimation can bene t from postselecting θ the output data before postprocessing. Postselection can raise the parameter that governs a physical transformation. Fisher information per final measurement or postprocessing Optimized prepare-measure experiment: An input system event. Postselection can also raise the rate of information per final undergoes the partially unknown transformation, after which measurement in a quantum setting. But classical postselection is the system is measured. Both the input system and the intuitive, whereas an intense discussion surrounds postselected measurement are chosen to provide the largest possible Fisher quantum experiments17–29. The ontological nature of post- information. Postselected prepare-measure experiment: An input system selected quantum states, and the extent to which they exhibit fi nonclassical behavior, is subject to an ongoing debate. Particular undergoes, rst, the partially unknown transformation and, interest has been aimed at pre- and postselected averages of second, a postselection measurement. Conditioned on the “ ” ’ postselection’s yielding the desired outcome, the system under- observables. These weak values can lie outside an observable s fi eigenspectrum when measured via a weak coupling to a pointer goes an information-optimized nal measurement. particle17,30. Such values offer metrological advantages in esti- In quantum parameter estimation, a quantum state is mations of weak-coupling strengths19,24,31–36. measured to reveal information about an unknown parameter In this article, we go beyond this restrictive setting and ask, can encoded in the state. We now compare, in this quantum setting, postselection provide a nonclassical advantage in general quan- the Fisher-information values generated from the two metrolo- tum parameter-estimation experiments? We conclude that it can. gical procedures described above. Consider a quantum experi- ^ρ ^ θ ^ρ ^ y θ ^ρ We study metrology experiments for estimating an unknown ment that outputs a state θ ¼ Uð Þ 0U ð Þ, where 0 is the transformation parameter whose final measurement or post- input state and U^ ðθÞ represents a unitary evolution set by θ. The processing incurs an experimental cost37,38. Postselection allows quantum Fisher information is defined as the Fisher information the experiment to incur that cost only when the postselected maximized over all possible generalized measurements7,13,39,40: measurement’s result reveals that the final measurement’s Fisher hi 2 information will be sufficiently large. We express the Fisher θ ^ρ ^ρ Λ^ : I Qð j θÞ¼Tr θ ^ρθ ð2Þ information in terms of a quasiprobability distribution. Quantum negativity in this distribution enables postselection to increase the Λ^ fi ^ρθ is the symmetric logarithmic derivative, implicitly de ned by Fisher information above the values available from standard 1 ^ ^ 12 ∂θ^ρθ ¼ ðΛ^ρ ^ρθ þ ^ρθΛ^ρ Þ . input-and-measurement-optimized experiments. Such an anom- 2 θ θ If ^ρ is pure, such that ^ρ ¼ jiΨθ hjΨθ , the quantum Fisher alous Fisher information can improve the rate of information θ θ information can be written as32,41 gain to experimental cost, offering a genuine quantum advantage in metrology. We show that, within a commuting theory, a theory θ ^ρ Ψ_ Ψ_ Ψ_ Ψ 2; I Qð j θÞ¼4h θj θiÀ4jh θj θij ð3Þ in which observables commute classically, postselection can _ improve information-cost rates no more than a strategy that uses where Ψθ ∂θjiΨθ . 2 NATURE COMMUNICATIONS | (2020) 11:3775 | https://doi.org/10.1038/s41467-020-17559-w | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-17559-w ARTICLE 1–F distribution. Many classes of quasiprobability distributions exist. 0 U( ) The most famous is the Wigner function43. Such a distribution F ps satisfies some, but not all, of Kolmogorov’s axioms for probability distributions44: the entries sum to unity, and marginalizing over Fig. 2 Preparation of postselected quantum state. First, an input
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