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arXiv:2108.09220 · Physical Review A

Saturating quantum advantages in postselected metrology with the positive Kirkwood-Dirac distribution

Sourav Das, Subhrajit Modak, Manabendra Nath Bera

Published Apr 11, 2023• 18 citations

Abstract

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.

DOI: 10.1103/physreva.107.042413

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

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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

Achieving metrological precision limits through postselection

G. B. Alves, Arthur Pimentel, Malena O. Hor-Meyll +3 more

Postselection strategies have been proposed with the aim of amplifying weak signals, which may help to overcome detection thresholds associated with technical noise in high-precision measurements. Here we use an optical setup to experimentally explore two different postselection protocols for the estimation of a small parameter: a weak-value amplification procedure and an alternative method that does not provide amplification but nonetheless is shown to be more robust for the sake of parameter estimation. Each technique leads approximately to the saturation of quantum limits for the estimation precision, expressed by the Cram\'er-Rao bound. For both situations, we show that parameter estimation is improved when the postselection statistics are considered together with the measurement device.

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Entanglement-Assisted Weak Value Amplification

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Large weak values have been used to amplify the sensitivity of a linear response signal for detecting changes in a small parameter, which has also enabled a simple method for precise parameter estimation. However, producing a large weak value requires a low postselection probability for an ancilla degree of freedom, which limits the utility of the technique. We propose an improvement to this method that uses entanglement to increase the efficiency. We show that by entangling and postselecting n ancillas, the postselection probability can be increased by a factor of n while keeping the weak value fixed (compared to n uncorrelated attempts with one ancilla), which is the optimal scaling with n that is expected from quantum metrology. Furthermore, we show the surprising result that the quantum Fisher information about the detected parameter can be almost entirely preserved in the postselected state, which allows the sensitive estimation to approximately saturate the relevant quantum Cramér-Rao bound. To illustrate this protocol we provide simple quantum circuits that can be implemented using current experimental realizations of three entangled qubits.

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

Sourav Das, Subhrajit Modak, Manabendra Nath Bera

In Das et al. [Phys. Rev. A 107, 042413 (2023)], we showed that Kirkwood-Dirac negativity is not necessary to achieve anomalous quantum Fisher information. Arvidsson-Shukur et al. [preceding Comment, Phys. Rev. A (to be published)] claim that one of our conclusions is wrong. Here, we argue that the claim is unfounded and incorrect and conclusions in Das et al. stand correct when the standard form of the Kirkwood-Dirac distribution is investigated. The confusion arises due to the choice of the distribution under consideration. We clarify this issue along with an interpretation of nonclassicality attached to the positivity of Kirkwood-Dirac distribution.

Physical Review A•Jun 21, 2024•0 citations

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.

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.

Postselection strategies have been proposed with the aim of amplifying weak signals, which may help to overcome detection thresholds associated with technical noise in high-precision measurements. Here we use an optical setup to experimentally explore two different postselection protocols for the estimation of a small parameter: a weak-value amplification procedure and an alternative method that does not provide amplification but nonetheless is shown to be more robust for the sake of parameter estimation. Each technique leads approximately to the saturation of quantum limits for the estimation precision, expressed by the Cram\'er-Rao bound. For both situations, we show that parameter estimation is improved when the postselection statistics are considered together with the measurement device.

Large weak values have been used to amplify the sensitivity of a linear response signal for detecting changes in a small parameter, which has also enabled a simple method for precise parameter estimation. However, producing a large weak value requires a low postselection probability for an ancilla degree of freedom, which limits the utility of the technique. We propose an improvement to this method that uses entanglement to increase the efficiency. We show that by entangling and postselecting n ancillas, the postselection probability can be increased by a factor of n while keeping the weak value fixed (compared to n uncorrelated attempts with one ancilla), which is the optimal scaling with n that is expected from quantum metrology. Furthermore, we show the surprising result that the quantum Fisher information about the detected parameter can be almost entirely preserved in the postselected state, which allows the sensitive estimation to approximately saturate the relevant quantum Cramér-Rao bound. To illustrate this protocol we provide simple quantum circuits that can be implemented using current experimental realizations of three entangled qubits.

In Das et al. [Phys. Rev. A 107, 042413 (2023)], we showed that Kirkwood-Dirac negativity is not necessary to achieve anomalous quantum Fisher information. Arvidsson-Shukur et al. [preceding Comment, Phys. Rev. A (to be published)] claim that one of our conclusions is wrong. Here, we argue that the claim is unfounded and incorrect and conclusions in Das et al. stand correct when the standard form of the Kirkwood-Dirac distribution is investigated. The confusion arises due to the choice of the distribution under consideration. We clarify this issue along with an interpretation of nonclassicality attached to the positivity of Kirkwood-Dirac distribution.