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 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 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 ðθÞފ ¼ ½∂θp ðθފ : 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 ފ. 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 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 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 quantum ^ iθA^ the eigenvalues of every observable except one yields a probability state ^ρ undergoes a unitary transformation UðθÞ¼eÀ : ^ρ ! ^ρ . Second, 0 0 θ distribution over the remaining observable’s eigenvalues. A qua- the quantum state is subject to a projective postselective measurement ^ ^ siprobability distribution can, however, have negative or nonreal fF;^1 À Fg. The postselection is such that if the outcome related to the ^F values. Such values signal nonclassical physics in, for example, operator happens, then the quantum state is not destroyed. The 2,6,45–53 ps ^ ^ ^ and . experiment outputs renormalized states ^ρ ¼ F^ρ F=TrðF^ρ Þ. θ θ θ A cousin of the Wigner function is the Kirkwood–Dirac quasiprobability distribution4–6. This distribution, which has been referred to by several names across the literature, resembles We assume that the evolution can be represented in accordance ^ the Wigner function for continuous systems. Unlike the Wigner with Stone’s theorem42,byU^ ðθÞeÀiAθ, where A^ is a Hermitian functions, however, the Kirkwood–Dirac distribution is well- operator. We assume that A^ is not totally degenerate: If all the A^ defined for discrete systems, even qubits. The Kirkwood–Dirac eigenvalues were identical, U^ ðθÞ would not imprint θ onto the distribution has been used in the study of weak-value fi 6,48,54–57 6,52,53,58 state in a relative phase. For a pure state, the quantum Fisher ampli cation , information scrambling , and 59–61 θ ^ρ = ^ 7 direct measurements of quantum wavefunctions . Moreover, information equals I Qð j θÞ 4VarðAÞ^ρ . Maximizing Eq. (1) θ ^ρ 0 θ ^ρ negative and nonreal values of the distribution have been linked over all measurements gives I Qð j θÞ. Similarly, I Qð j θÞ can be to nonclassical phenomena6,48,52,53. We cast the quantum Fisher ^ ^θ maximized over all input states. For a given unitary UðθÞ¼eÀiA , information for a postselected prepare-measure experiment in the maximum quantum Fisher information is terms of a doubly extended Kirkwood–Dirac quasiprobability ÈÉ no distribution6. (The modifier “doubly extended” comes from the 2 max^ρ I ðθj^ρθÞ ¼ 4max^ρ Var ðA^Þρ ¼ðΔaÞ ; ð4Þ 0 Q 0 ^0 experiment in which one would measure the distribution: One would prepare ^ρ, sequentially measure two observables weakly, where Δa is the difference between the maximum and minimum and measure one observable strongly. The number of weak eigenvalues of A^7. (The information-optimal input state is a pure measurements equals the degree of the extension6.) We employ state in an equal superposition of one eigenvector associated with this distribution due to its usefulness as a mathematical tool: This the smallest eigenvalue and one associated with the largest.) To distribution enables the proof that, in the presence of non- summarize, in an optimized quantum prepare-measure experi- commuting observables, postselection can give a metrological ment, the quantum Fisher information is (Δa)2. protocol a nonclassical advantage. ^ ^ We now find an expression for the quantum Fisher informa- Our distribution is defined in terms of eigenbases of A and F. tion in a postselected prepare-measure experiment. A projective Other quasiprobability distributions are defined in terms of bases postselection occurs after U^ ðθÞ but before the final measurement. independent of the experiment. For example, the Wigner function Figure 2 shows such a . The renormalized is often defined in the bases of the quadrature of the electric field quantum state that passes the postselection is or the position and momentum bases. However, basis- qffiffiffiffiffiffi ps ps ps independent distributions can be problematic in the hunt for Ψ  ψ = p , where we have defined an unnormalized θ θ θ nonclassicality2,49. Careful application, here, of the extended ps ^ – state ψθ  FjiΨθ and the postselection probability Kirkwood Dirac distribution ties its nonclassical values to the P fi ps ^^ρ ^ρ ^ θ ^ρ ^ y θ ^ operational speci cs of the experiment. pθ  TrðF θÞ. As before, θ ¼ Uð Þ 0U ð Þ. F ¼ f 2F ps jif hjf fi ps To begin, we de ne the quasiprobability distribution of an is the postselecting projection operator, and F is a set of arbitrary quantum state ^ρ: orthonormal basis states allowed by the postselection. Finally, the postselected state undergoes an information-optimal qffiffiffiffiffiffi ^ρ ^ρ 0 0 : measurement. q ; 0; hf jaihja jiha a jf i ð6Þ ps ps ps a a f When Ψθ  ψθ = pθ is substituted into Eq. (3), the ps derivatives of pθ cancel, such that Here, fjigf , fjiga , and fjiga0 are bases for the Hilbert space on θ Ψps ψ_ ps ψ_ ps 1 ψ_ ps ψps 2 1 : which ^ρ is defined. We can expand ^ρ59,60 as I Qð j θ Þ¼4h θ j θ i ps À 4jh θ j θ ij ps 2 ð5Þ pθ ðpθ Þ X Equation (5) gives the quantum Fisher information available from ^ jia hjf ^ρ ρ ¼ q 0 : θ Ψps h j i a;a ;f ð7Þ a quantum state after its postselection. Unsurprisingly, I Qð j θ Þ a;a0;f f a θ ^ρ ps ≤ can exceed I Qð j θÞ, since pθ 1. Also classical systems can achieve such postselected information amplification (see Fig. 1). ps If any hf jai = 0, we perturb one of the bases infinitesimally while Unlike in the classical case, however, I ðθjΨ Þ can also exceed Q θ preserving its orthonormality. the Fisher information of an optimized prepare-measure experi- Let fjiga ¼fjiga0 denote an eigenbasis of A^, and let fjigf ment, (Δa)2. We show how below. denote an eigenbasis of F^. The reason for introducing a doubly extended distribution, instead of the standard Kirkwood–Dirac Quasiprobability representation. In , our ^ρ ^ρ θ Ψps knowledge of a point particle can be described by a probability distribution qa;f hf jaihja jif , is that I Qð j θ Þ can be distribution for the particle’s position, x, and momentum, k: p expressed most concisely, naturally, and physically meaningfully ^ρ (x, k). In quantum mechanics, position and momentum do not θ in terms of qa;a0;f . Later, we shall see how the nonclassical entries commute, and a state cannot generally be represented by a joint ^ρ ^ρ in q 0 and q are related. We now express the postselected probability distribution over observables’ eigenvalues. A quantum a;a ;f a;f state can, however, be represented by a quasiprobability quantum Fisher information (Eq. (5)) in terms of the

NATURE COMMUNICATIONS | (2020) 11:3775 | https://doi.org/10.1038/s41467-020-17559-w | www.nature.com/naturecommunications 3 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-17559-w

^ρ θ ^ρ ^ ^ fi quasiprobability values q ; 0; (see Supplementary Note 1). pairwise noncommutation of θ, A, and F is insuf cient to enable a a f ps 2 anomalous values of I ðθjΨθ Þ. For example, noncommutation ^ρ ^ρ Q X θ X θ – q ; 0; q ; 0; could lead to a nonreal Kirkwood Dirac distribution without any θ Ψps a a f 0 a a f ; I Qð j θ Þ¼4 ps aa À 4 ps a negative real components. Such a distribution cannot improve pθ pθ ps ; 0; ps ; 0; ps θ Ψ a a f 2F a a f 2F I Qð j θ Þ beyond classical values. Furthermore, the presence or absence of commutation is a binary measure. In contrast, how ð8Þ θ Ψps much postselection improves I Qð j θ Þ depends on how much 0 where a and a denote the eigenvalues associated with jia and ^ρθ ps negativity q 0 =p has. We build on this observation, and jia0 , respectively. (We have suppressed degeneracy parameters γ a;a ;f θ in our notation for the states, e.g., jia; γ  jia .) Equation (8) propose two experiments that yield anomalous Fisher- ^ρθ ps information values, in Supplementary Notes 3 and 4. (It remains contains a conditional quasiprobability distribution, q 0 =p .If a;a ;f θ an open question to investigate the relationship between ^ ^ A commutes with F, as they do classically, then they share an Kirkwood–Dirac negativity in other metrology protocols with ^ρ θ = ps ; noncommuting operators, e.g., ref. 62.) eigenbasis for which qa;a0;f pθ 2½0 1Š, and the postselected θ Ψps ≤ Δ 2 As promised, we now address the relation between nonclassical quantum Fisher information is bounded as I Qð j θ Þ ð aÞ : ^ρ ^ρ Theorem 1 In a classically commuting theory, no postselected entries in qa;a0;f and nonclassical entries in qa;f . For pure prepare-measure experiment can generate more Fisher informa- states ^ρ ¼ jiΨ hjΨ , the doubly extended quasiprobability distribu- tion than the optimized prepare-measure experiment. tion can be expressed time symmetrically in terms of the standard à Proof: We upper-bound the right-hand side of Eq. (8). First, if – 4–6,51–53 ^ρ 1 ^ρ ^ρ Kirkwood Dirac distribution : qa;a0;f ¼ p qa;f ðqa0;f Þ , 0 ^ ^ f fjiga ¼fjiga ¼fjigf is a eigenbasis shared by A and F, Eq. (6) ^ρ ^ρ ≡∣〈 ∣Ψ〉∣2 20,63 simplifies to a probability distribution: where qa;f ¼hf jaihja jif and pf f . (See refs. for ^ρ discussions about time-symmetric interpretations of quantum θ ^ρ 0 0 ; ; ^ρ qa;a0;f ¼ hja θji½a ji¼f ja iŠ½ji¼a jiŠ2½f 0 1Š ð9Þ mechanics.) Therefore, a negative qa;a0;f implies negative or where [X] is the Iverson bracket, which equals 1 if X is true and ^ρ ^ρ ^ρ nonreal values of qa;f . Similarly, a negative qa;a0;f implies a θ = ps ps 17 equals 0 otherwise. Second, summing qa;a0;f pθ over f 2F ,we negative or nonreal 〈f ∣a〉〈a∣Ψ〉/〈f ∣Ψ〉 , which find possesses interesting ontological features (see below). Thus, an X ^ρ anomalous Fisher information is closely related to a negative or θ = ps ^ρ 0 0 ^ = ps: qa;a0;f pθ ¼ hja θjia hja Fajipθ ð10Þ nonreal weak value. Had we weakly measured the observable f 2F ps jia hja of ^ρθ with a qubit or Gaussian pointer particle before the fi By the eigenbasis shared by A^ and F^, the sum simplifies to postselection, and had we used a ne-grained postselection ^ ^ ps ^ 0 0 ps f À ; : 2F g hja ^ρθFaji½jia ¼ jiŠa =pθ . We can thus rewrite Eq. (8): 1 F jif hjf f , the weak measurement would have yielded a weak value outside the eigenspectrum of jia hja . It has X ^ 0 0 hja ^ρθFaji½ji¼a jiŠa I ðθjΨpsÞ¼ 4 aa0 been shown that such an anomalous weak value proves that Q θ ps quantum mechanics, unlike classical mechanics, is contextual: a;a0 pθ quantum outcome probabilities can depend on more than a 2 X ^ρ ^ 0 0 unique set of underlying physical states24,35,64.If^ρ had hja θFaji½ji¼a jiŠa θ À 4 ps a ð11Þ undergone the aforementioned weak measurement, instead of ; 0 pθ a a ! the postselected prepare-measure experiment, the weak measure- X X 2 ment’s result would have signaled quantum contextuality. 2 ; ¼ 4 qaa À 4 qaa Consequently, a counterfactual connects an anomalous Fisher a a information and quantum contextuality. While counterfactuals fi create no problems in classical physics, they can lead to logical where we haveP de ned the probabilities – ^ρ ^ = ps ^ρ = ps paradoxes in quantum mechanics64 67. Hence our counter- qa  hja θFajipθ ¼ f 2F ps hja θji½a jif ¼ jiŠa pθ . ’ Apart from the multiplicative factor of 4, Eq. (11) is in the form factual s implication for the ontological relation between an of a variance with respect to the observable’s eigenvalues a. Thus, anomalous Fisher information and contextuality offers an opportunity for future investigation. Eq. (11) is maximized when q ¼ q ¼ 1: amin amax 2 θ Ψps Δ 2: maxfI Qð j θ Þg ¼ ð aÞ Improved metrology via postselection. In every real experiment, fq g ð12Þ a the preparation and final measurement have costs, which we This Fisher-information bound must be independent of our denote CP and CM, respectively. For example, a particle- choice of eigenbases of A^ and F^. In summary, if A^ commutes with number detector’s dead time, the time needed to reset after a ^ρ 68 ^ θ = ps detection, associates a temporal cost with measurements . F, then all q ; 0; pθ can be expressed as real and nonnegative, a a f Reference37 concerns a two-level atom in a noisy environment. θ Ψps ≤ Δ 2 and I Qð j θ Þ ð aÞ . Liuzzo et al. detail the tradeoff between frequency estimation’s In contrast, if the quasiprobability distribution contains time and energy costs. Standard quantum-metrology techniques, negative values, the postselected quantum Fisher information they show, do not necessarily improve metrology, if the experi- θ Ψps Δ 2 can violate the bound: I Qð j θ Þ > ð aÞ . ment’s energy is capped. Also, the cost of postprocessing can be Theorem 2 An anomalous postselected Fisher information incorporated into CM. (In an experiment, these costs can implies that the quantum Fisher information cannot be expressed be multivariate functions that reflect the resources and constraints. in terms of a nonnegative doubly extended Kirkwood–Dirac Such a function could combine a detector’s dead time with quasiprobability distribution. the monetary cost of liquid helium and a graduate student’s Proof: See Supplementary Note 2 for a proof. salary. However, presenting the costs in a general form benefits (The theorem’s converse is not generally true.) This inability to this platform-independent work.) We define the information-cost ^ ^ θ : θ = θ = express implies that A fails to commute with F. However, rate as Rð Þ ¼ NIð Þ ðNCP þ NCMÞ¼Ið Þ ðCP þCMÞ.Ifour

4 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 experiment conditions the execution of the final measurement on postselection can improve information-cost rates, as long as ps ps — fi successful postselection of a fraction pθ of the states, we include a Cps < ð1 À pθ ÞCM a signi cantly weaker requirement than fi ’ cost of postselection, Cps.Wede ne the postselected experiment s CM CP þCps. ps θ : ps ps θ = information-cost rate as R ð Þ ¼ Npθ I ð Þ ðNCP þ NCps þ ps ps ps θ = ps ps θ Discussion Npθ CMÞ¼pθ I ð Þ ðCP þCps þ pθ CMÞ, where I ð Þ is the Fisher information conditioned on successful postselection. Gen- From a practical perspective, our results highlight an important eralizing the following arguments to preparation and measure- quantum asset for parameter-estimation experiments with expensive final measurements. In some scenarios, the postselec- ment costs that differ between the postselected and ’ fi ’ nonpostselected experiments is straightforward. tion s costs exceed the nal measurement s costs, as an unsuc- cessful postselection might require fast feedforward to block In classical experiments, postselection can improve the fi information-cost rate. See Fig. 1 for an example. But can the nal measurement. But in single-particle experiments, the postselection improve the information-cost rate in a classical postselection can be virtually free and, indeed, unavoidable: experiment with information-optimized inputs? Theorem 1 an unsuccessful postselection can destroy the particle, precluding ps θ ≤ θ the triggering of the final measurement’s detection apparatus80. answered this question in the negative. I ð Þ maxfIð Þg in fi every classical experiment. The maximization is over all Thus, current single-particle metrology could bene t from physically accessible inputs and final measurements. A direct postselected improvements of the Fisher information. A implication is that RpsðθÞ ≤ maxfRðθÞg. photonic experimental test of our results is currently under θ Ψps investigation. In quantum mechanics, I Qð j θ Þ can exceed θ ^ρ Δ 2 From a fundamental perspective, our results highlight the max^ρ fI Qð j θÞg ¼ ð aÞ . This result would be impossible 0 strangeness of quantum mechanics as a noncommuting theory. classically. Anomalous Fisher-information values require quan- Classically, an increase of the Fisher information via postselection – tum negativity in the doubly extended Kirkwood Dirac distribu- can be understood as the a posteriori selection of a better input tion. Consequently, even compared to quantum experiments with distribution. But it is nonintuitive that quantum-mechanical optimized input states, postselection can raise information-cost postselection can enable a quantum state to carry more Fisher ps θ θ rates beyond classically possible rates: R ð Þ> maxfRð Þg. This information than the best possible input state could. Further- result generalizes the metrological advantages observed in the more, it is surprising that noncommutation can be proved to measurements of weak couplings, which also require noncom- underlie the metrological advantage: Other nonclassical phe- 69–77 muting operators. References concern metrology that nomena, such as entanglement and discord, could be expected to involves weak measurements of the following form. The primary underlie a given nonclassical advantage; noncommutation does system S and the pointer P begin in a pure product state not guarantee a metrological advantage; and the proof turns out Ψ Ψ ji S jiP ; the coupling Hamiltonian is a product to involve considerable mathematical footwork. The optimized ^ ^ ^ θ H ¼ AS AP; the unknown coupling strength is small; and Cramér–Rao bound, obtained frompffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eq. (4), can be written in the just the system is postselected. Our results govern arbitrary input θ Δ ≥ 7 form of an uncertainty relation: Varð eÞð aÞ 1 . Our results states, arbitrary Hamiltonians (that satisfy Stone’s theorem), highlight the probabilistic possibility of violating this bound. arbitrarily large coupling strengths θ, and arbitrary projective More generally, the information-cost rate’s ability to violate a postselections. Our result shows that postselection can improve classical bound leverages negativity, a nonclassical resource in quantum parameter estimation in experiments where the final , for metrological advantage. measurement’s cost outweighs the combined costs of state preparation and postselection: CM CP þCps. Earlier works Data availability identified that the Fisher information from nonrenormalized The authors declare that all data supporting the findings of this study are available within trials that succeed in the postselection cannot exceed the Fisher the paper and its supplementary information files. information averaged over all trials, including the trials in which the postselection fails78,79. (Reference 41 considered squeezed Received: 20 November 2019; Accepted: 8 July 2020; coherent states as metrological probes in specific weak- measurement experiments. It is shown that postselection can improve the signal-to-noise ratio, irrespectively of whether the analysis includes the failed trials. However, this work concerned nonpostselected experiments in which only the probe state was References measured. Had it been possible to successfully measure also the 1. Lütkenhaus, N. & Barnett, S. M. Nonclassical effects in phase space. Phys. Rev. target system, the advantage would have disappeared.) In A 51, 3340–3342 (1995). accordance with practical metrology, not only the Fisher 2. Spekkens, R. W. Negativity and contextuality are equivalent notions of information, but also measurements’ experimental costs, underlie nonclassicality. Phys. Rev. Lett. 101, 020401 (2008). 3. Ferrie, C. & Emerson, J. Frame representations of quantum mechanics and the our results. necessity of negativity in quasi-probability representations. J. Phys. A: Math. θ Ψps Δ 2 So far, we have shown that I Qð j θ Þ can exceed ( a) . But Theor. 41, 352001 (2008). θ Ψps 4. Kirkwood, J. G. Quantum statistics of almost classical assemblies. Phys. Rev. how large can I Qð j θ Þ grow? In Supplementary Note 3, we ^ ≥ 44,31–37 (1933). show that, if the generator A has M 3 not-all-identical 5. Dirac, P. A. M. On the analogy between classical and quantum mechanics. θ Ψps eigenvalues, there is no upper bound on I Qð j θ Þ.IfCP and Rev. Mod. Phys. 17, 195–199 (1945). 6. Yunger Halpern, N., Swingle, B. & Dressel, J. Quasiprobability behind the out- Cps are negligible compared with CM, then there is no theoretical ps θ of-time-ordered correlator. Phys. Rev. A 97, 042105 (2018). cap on how large R ( ) can grow. In general, when 7. Giovannetti, V., Lloyd, S. & Maccone, L. Advances in quantum metrology. θ Ψps ps ´ θ Ψps Δ 2 – I Qð j θ Þ!1, pθ I Qð j θ Þ < ð aÞ , such that information Nat. Photonics 5, 222 229 (2011). is lost in the events discarded by postselection. But if A^ has 8. Giovannetti, V., Lloyd, S. & Maccone, L. Quantum metrology. Phys. Rev. Lett. 96, 010401 (2006). doubly degenerate minimum and maximum eigenvalues, 9. Krischek, R. et al. Useful multiparticle entanglement and sub-shot-noise ps ´ θ Ψps Δ 2 θ Ψps pθ I Qð j θ Þ can approach ( a) while I Qð j θ Þ approaches sensitivity in experimental phase estimation. Phys. Rev. Lett. 107, 080504 infinity (see Supplementary Note 4). In such a scenario, (2011).

NATURE COMMUNICATIONS | (2020) 11:3775 | https://doi.org/10.1038/s41467-020-17559-w | www.nature.com/naturecommunications 5 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-17559-w

10. Demkowicz-Dobrzański, R. & Maccone, L. Using entanglement against noise 45. Ferrie, C. Quasi-probability representations of quantum theory with in quantum metrology. Phys. Rev. Lett. 113, 250801 (2014). applications to science. Rep. Prog. Phys. 74, 116001 11. Maccone, L. Intuitive reason for the usefulness of entanglement in quantum (2011). metrology. Phys. Rev. A 88, 042109 (2013). 46. Kofman, A. G., Ashhab, S. & Nori, F. Nonperturbative theory of weak pre- 12. Helstrom, C. W. Quantum Detection and Estimation Theory. 1st edn and post-selected measurements. Phys. Rep. 520,43–133 (2012). (Academic Press, New York, 1976). 47. Howard, M., Wallman, J., Veitch, V. & Emerson, J. Contextuality supplies the 13. Braunstein, S. L. & Caves, C. M. Statistical distance and the geometry of magic for quantum computation. Nature 510, 351–355 (2014). quantum states. Phys. Rev. Lett. 72, 3439–3443 (1994). 48. Dressel, J. Weak values as interference phenomena. Phys. Rev. A 91, 032116 14. Cover, T. M. & Thomas, J. A. Elements of Information Theory. 2nd edn (John (2015). Wiley and Sons Inc., Hoboken, 2006). 49. Delfosse, N., Allard Guerin, P., Bian, J. & Raussendorf, R. Wigner function 15. Cramér, H. Mathematical Methods of Statistics (PMS-9), Vol. 9 (Princeton negativity and contextuality in quantum computation on rebits. Phys. Rev. X University Press, 2016). 5, 021003 (2015). 16. Rao, C. R. in Breakthroughs in Statistics 235–247 (Springer, 1992). 50. Delfosse, N., Okay, C., Bermejo-Vega, J., Browne, D. E. & Raussendorf, R. 17. Aharonov, Y., Albert, D. Z. & Vaidman, L. How the result of a measurement Equivalence between contextuality and negativity of the wigner function for of a component of the of a spin-1/2 particle can turn out to be 100. Phys. qudits. N. J. Phys. 19, 123024 (2017). Rev. Lett. 60, 1351–1354 (1988). 51. Yunger Halpern, N. Jarzynski-like equality for the out-of-time-ordered 18. Leifer, M. S. & Spekkens, R. W. Pre- and post-selection paradoxes and correlator. Phys. Rev. A 95, 012120 (2017). contextuality in quantum mechanics. Phys. Rev. Lett. 95, 200405 (2005). 52. Halpern, N. Y., Bartolotta, A. & Pollack, J. Entropic uncertainty relations for 19. Tollaksen, J. Pre- and post-selection, weak values and contextuality. J. Phys. A: quantum information scrambling. Commun. Phys. 2,1–12 (2019). Math. Theor. 40, 9033–9066 (2007). 53. González Alonso, J., Yunger Halpern, N. & Dressel, J. Out-of-time-ordered- 20. Aharonov, Y. & Vaidman, L. in Time in Quantum Mechanics 235–247 correlator quasiprobabilities robustly witness scrambling. Phys. Rev. Lett. 122, (Springer, 2008). 040404 (2019). 21. Dressel, J., Agarwal, S. & Jordan, A. N. Contextual values of observables in 54. Steinberg, A. M. Conditional probabilities in quantum theory and the quantum measurements. Phys. Rev. Lett. 104, 240401 (2010). tunneling-time controversy. Phys. Rev. A 52,32–42 (1995). 22. Vaidman, L. Past of a quantum particle. Phys. Rev. A 87, 052104 (2013). 55. Johansen, L. M. Quantum theory of successive projective measurements. Phys. 23. Ferrie, C. & Combes, J. How the result of a single coin toss can turn out to be Rev. A 76, 012119 (2007). 100 heads. Phys. Rev. Lett. 113, 120404 (2014). 56. Hofmann, H. F. Complex joint probabilities as expressions of reversible 24. Pusey, M. F. Anomalous weak values are proofs of contextuality. Phys. Rev. transformations in quantum mechanics. N. J. Phys. 14, 043031 (2012). Lett. 113, 200401 (2014). 57. Piacentini, F. et al. Measuring incompatible observables by exploiting 25. Pusey, M. F. & Leifer, M. S. Logical pre- and post-selection paradoxes are sequential weak values. Phys. Rev. Lett. 117, 170402 (2016). proofs of contextuality, in Proceedings of the 12th International Workshop on 58. Mohseninia, R., Alonso, J. R. & Dressel, J. Optimizing measurement strengths Quantum Physics and Logic, Vol. 195, 295–306 (Open Publishing Association, for qubit quasiprobabilities behind out-of-time-ordered correlators. Phys. Rev. 2015). A 100, 062336 (2019). 26. Arvidsson-Shukur, D. R. M. & Barnes, C. H. W. Quantum counterfactual 59. Lundeen, J. S., Sutherland, B., Patel, A., Stewart, C. & Bamber, C. communication without a weak trace. Phys. Rev. A 94, 062303 (2016). Direct measurement of the quantum wavefunction. Nature 474, 188–191 27. Arvidsson-Shukur, D. R. M., Gottfries, A. N. O. & Barnes, C. H. W. Evaluation (2011). of counterfactuality in counterfactual communication protocols. Phys. Rev. A 60. Lundeen, J. S. & Bamber, C. Procedure for direct measurement of general 96, 062316 (2017). quantum states using weak measurement. Phys. Rev. Lett. 108, 070402 (2012). 28. Arvidsson-Shukur, D. R. M. & Barnes, C. H. W. Postselection and 61. Thekkadath, G. S. et al. Direct measurement of the of a counterfactual communication. Phys. Rev. A 99, 060102 (2019). quantum system. Phys. Rev. Lett. 117, 120401 (2016). 29. Cimini, V., Gianani, I., Piacentini, F., Degiovanni, I. P. & Barbieri, M. 62. Sun, L. et al. Exponentially enhanced quantum metrology using resources of Anomalous values, fisher information, and contextuality, in generalized linear complexity. Preprint at http://arXiv.org/abs/2004.01216 (2020). quantum measurements. Quantum Sci. Technol. 5, 025007 (2020). 63. Leifer, M. S. & Pusey, M. F. Is a time symmetric interpretation of quantum 30. Duck, I. M., Stevenson, P. M. & Sudarshan, E. C. G. The sense in which a theory possible without retrocausality? Proc. R. Soc. A: Math., Phys. Eng. Sci. “weak measurement” of a spin-1/2 particle’s spin component yields a value 473, 20160607 (2017). 100. Phys. Rev. D. 40, 2112–2117 (1989). 64. Spekkens, R. W. Contextuality for preparations, transformations, and unsharp 31. Dressel, J., Malik, M., Miatto, F. M., Jordan, A. N. & Boyd, R. W. Colloquium: measurements. Phys. Rev. A 71, 052108 (2005). understanding quantum weak values: Basics and applications. Rev. Mod. Phys. 65. Kochen, S. & Specker, E. P. in The Logico-algebraic Approach to Quantum 86, 307–316 (2014). Mechanics 293–328 (Springer, 1975). 32. Pang, S., Dressel, J. & Brun, T. A. Entanglement-assisted weak value 66. Hardy, L. Quantum mechanics, local realistic theories, and lorentz-invariant amplification. Phys. Rev. Lett. 113, 030401 (2014). realistic theories. Phys. Rev. Lett. 68, 2981–2984 (1992). 33. Jordan, A. N., Martínez-Rincón, J. & Howell, J. C. Technical advantages for 67. Penrose, R. Shadows of the Mind: A Search for the Missing Science of weak-value amplification: When less is more. Phys. Rev. X 4, 011031 (2014). Consciousness 240 (Oxford University Press, 1994). 34. Harris, J., Boyd, R. W. & Lundeen, J. S. Weak value amplification can 68. Greganti, C. et al. Tuning single-photon sources for telecom multi-photon outperform conventional measurement in the presence of detector saturation. experiments. Opt. Express 26, 3286–3302 (2018). Phys. Rev. Lett. 118, 070802 (2017). 69. Hosten, O. & Kwiat, P. Observation of the spin hall effect of light via weak 35. Kunjwal, R., Lostaglio, M. & Pusey, M. F. Anomalous weak values and measurements. Science 319, 787–790 (2008). contextuality: robustness, tightness, and imaginary parts. Phys. Rev. A 100, 70. Starling, D. J., Dixon, P. B., Jordan, A. N. & Howell, J. C. Optimizing the 042116 (2019). signal-to-noise ratio of a beam-deflection measurement with interferometric 36. Xu, L. et al. Approaching quantum-limited metrology with imperfect detectors weak values. Phys. Rev. A 80, 041803 (2009). by using weak-value amplification. Preprint at http://arXiv.org/abs/ 71. Brunner, N. & Simon, C. Measuring small longitudinal phase shifts: weak 2005.03629 (2020). measurements or standard interferometry? Phys. Rev. Lett. 105, 010405 37. Liuzzo-Scorpo, P. et al. Energy-efficient quantum frequency estimation. N. J. (2010). Phys. 20, 063009 (2018). 72. Starling, D. J., Dixon, P. B., Jordan, A. N. & Howell, J. C. Precision frequency 38. Lipka-Bartosik, P. & Demkowicz-Dobrzański, R. Thermodynamic work cost measurements with interferometric weak values. Phys. Rev. A 82, 063822 of quantum estimation protocols. J. Phys. A: Math. Theor. 51, 474001 (2018). (2010). 39. Fujiwara, A. & Nagaoka, H. Quantum fisher metric and estimation for pure 73. Egan, P. & Stone, J. A. Weak-value thermostat with 0.2 mk precision. Opt. state models. Phys. Lett. A 201, 119–124 (1995). Lett. 37, 4991–4993 (2012). 40. Petz, D. & Ghinea, C. Quantum Probability and Related Topics 261–281 74. Hofmann, H. F., Goggin, M. E., Almeida, M. P. & Barbieri, M. Estimation of a (World Scientific, 2011). quantum interaction parameter using weak measurements: theory and 41. Pang, S. & Brun, T. A. Improving the precision of weak measurements by experiment. Phys. Rev. A 86, 040102 (2012). postselection measurement. Phys. Rev. Lett. 115, 120401 (2015). 75. Magaña Loaiza, O. S., Mirhosseini, M., Rodenburg, B. & Boyd, R. W. 42. Stone, M. H. On one-parameter unitary groups in hilbert space. Ann. Math. Amplification of angular rotations using weak measurements. Phys. Rev. Lett. 33, 643–648 (1932). 112, 200401 (2014). 43. Wigner, E. On the quantum correction for thermodynamic equilibrium. Phys. 76. Lyons, K., Dressel, J., Jordan, A. N., Howell, J. C. & Kwiat, P. G. Power- Rev. 40, 749–759 (1932). recycled weak-value-based metrology. Phys. Rev. Lett. 114, 170801 (2015). 44. Kolmogorov, A. N. & Bharucha-Reid, A. T. Foundations of the Theory of 77. Martínez-Rincón, J., Mullarkey, C. A., Viza, G. I., Liu, W.-T. & Howell, J. C. Probability 2nd edn (Courier Dover Publications, 2018). Ultrasensitive inverse weak-value tilt meter. Opt. Lett. 42, 2479–2482 (2017).

6 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

78. Ferrie, C. & Combes, J. Weak value amplification is suboptimal for estimation Additional information and detection. Phys. Rev. Lett. 112, 040406 (2014). Supplementary information is available for this paper at https://doi.org/10.1038/s41467- 79. Combes, J., Ferrie, C., Jiang, Z. & Caves, C. M. Quantum limits on 020-17559-w. postselected, probabilistic quantum metrology. Phys. Rev. A 89, 052117 (2014). Correspondence and requests for materials should be addressed to D.R.M.A.-S. 80. Calafell, I. A. et al. Trace-free counterfactual communication with a nanophotonic processor. npj Quantum Inf. 5, 61 (2019). Peer review information Nature Communications thanks Rafal Demkowicz-Dobrzanski and the other anonymous reviewer(s) for their contribution to the peer review of this work.

Acknowledgements Reprints and permission information is available at http://www.nature.com/reprints The authors would like to thank Justin Dressel, Nicolas Delfosse, Matthew Pusey, Noah Lupu-Gladstein, Aharon Brodutch and Jan-Åke Larsson for useful discussions. D.R.M. Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in A.-S. acknowledges support from the EPSRC, the Sweden-America Foundation, Hitachi published maps and institutional affiliations. Ltd, Lars Hierta’s Memorial Foundation and Girton College. N.Y.H. was supported by an NSF grant for the Institute for Theoretical Atomic, Molecular, and Optical Physics at Harvard University and the Smithsonian Astrophysical Observatory. H.V.L. received funding from the European Union’s Horizon 2020 research and innovation programme Open Access This article is licensed under a Creative Commons under the Marie Skłodowska-Curie grant agreement No 642688. A.A.L. acknowledges Attribution 4.0 International License, which permits use, sharing, support from the EPSRC and Hitachi Ltd. S.L. was supported by NSF, AFOSR, and by adaptation, distribution and reproduction in any medium or format, as long as you give ARO under the Blue Sky Initiative. appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless Author contributions indicated otherwise in a credit line to the material. If material is not included in the ’ D.R.M.A.-S., N.Y.H., H.V.L., A.A.L., C.H.W.B., and S.L. contributed extensively to the article s Creative Commons license and your intended use is not permitted by statutory paper and to the writing of the manuscript. regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/ licenses/by/4.0/. Competing interests The authors declare no competing interests. © The Author(s) 2020

NATURE COMMUNICATIONS | (2020) 11:3775 | https://doi.org/10.1038/s41467-020-17559-w | www.nature.com/naturecommunications 7