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arXiv:1409.2567 · Physical Review Letters

Improving the Precision of Weak Measurements by Postselection Measurement

Shengshi Pang, Todd A. Brun

Published Sep 15, 2015• 119 citations
arXivHTMLPDF

Abstract

Postselected weak measurement is a useful protocol for amplifying weak physical effects. However, there has recently been controversy over whether it gives any advantage in precision. While it is now clear that retaining failed postselections can yield more Fisher information than discarding them, the advantage of postselection measurement itself still remains to be clarified. In this Letter, we address this problem by studying two widely used estimation strategies: averaging measurement results, and maximum likelihood estimation, respectively. For the first strategy, we find a surprising result that squeezed coherent states of the pointer can give postselected weak measurements a higher signal-to-noise ratio than standard ones while all standard coherent states cannot, which suggests that raising the precision of weak measurements by postselection calls for the presence of "nonclassicality" in the pointer states. For the second strategy, we show that the quantum Fisher information of postselected weak measurements is generally larger than that of standard weak measurements, even without using the failed postselection events, but the gap can be closed with a proper choice of system state.

DOI
10.1103/physrevlett.115.120401

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Postselected weak measurement is a useful protocol for amplifying weak physical effects. However, there has recently been controversy over whether it gives any advantage in precision. While it is now clear that retaining failed postselections can yield more Fisher information than discarding them, the advantage of postselection measurement itself still remains to be clarified. In this Letter, we address this problem by studying two widely used estimation strategies: averaging measurement results, and maximum likelihood estimation, respectively. For the first strategy, we find a surprising result that squeezed coherent states of the pointer can give postselected weak measurements a higher signal-to-noise ratio than standard ones while all standard coherent states cannot, which suggests that raising the precision of weak measurements by postselection calls for the presence of "nonclassicality" in the pointer states. For the second strategy, we show that the quantum Fisher information of postselected weak measurements is generally larger than that of standard weak measurements, even without using the failed postselection events, but the gap can be closed with a proper choice of system state.
We investigate, within the weak measurement theory, the advantages of nonclassical pointer states over semiclassical ones for coherent, squeezed vacuum, and Schr\"odinger cat states. These states are utilized as pointer states for the system operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}$ with property ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}}^{2}=\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{I}$, where $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{I}$ represents the identity operator. We calculate the ratio between the signal-to-noise ratio of nonpostselected and postselected weak measurements. The latter is used to find the quantum Fisher information for the above pointer states. The average shifts for those pointer states with arbitrary interaction strength are investigated in detail. One key result is that we find the postselected weak measurement scheme for nonclassical pointer states to be superior to semiclassical ones. This can improve the precision of the measurement process.
Postselected weak measurement has aroused broad interest for its distinctive ability to amplify small physical quantities. However, the low postselection efficiency to obtain a large weak value has been a big obstacle to its application in practice, since it can waste resources, and reduce the measurement precision. In this paper, we detail the entanglement-assisted postselected weak measurement protocol proposed in [Phys. Rev. Lett. 113, 030401 (2014)], and study its robustness against technical noises. We show that when the systems are properly entangled, postselection can be made dramatically more efficient, and as a consequence, the Fisher information can be increased to approximately saturate the well-known Heisenberg limit. We illustrate this protocol by a qubit example. Readout errors can greatly degrade the performance of postselected weak measurement schemes. We show that entanglement can significantly reduce two main detrimental effects of readout errors: inaccuracy in the measurement result, and loss of the Fisher information. We further compensate for readout errors by a majority vote scheme for postselection. With a proper threshold, almost no Fisher information will be lost. These results demonstrate the effectiveness of entanglement in protecting postselected weak measurement against readout errors.
Abstract Quantum metrology is being gradually studied for weak measurement systems. For weak measurement systems with a thermal state pointer, we find that in the displacement space corresponding to imaginary weak values, the maximal quantum Fisher information (QFI) after the postselection for an accepted event (conversely, the postselection for a rejected event) can attain a level of thermal fluctuation, without surpassing the total QFI, and further, that QFI which increases with increasing temperature can constantly improve measurement precision. These results are much better than those of weak measurement with a pure state (i.e. Gaussian state) pointer. On the other hand, in Kerr nonlinear interaction systems with weak measurement, and by using a thermal state pointer, we obtain in the phase space the accepted postselection and the postselected measurement both achieving the Heisenberg limit of quantum metrology, and showing that weak measurement with thermal states only obtain classical Fisher information (CFI) which increases with increasing temperature and achieves classical-enhanced scaling of N 2 . Moreover, weak measurement with thermal states has an advantage over that with coherent states or mixed states of light because generating these states with larger uncertainty is limited under current technology, but thermal states with larger uncertainly are very easy to achieve with increasing temperature in nature, such as for optical systems and optomechanical systems.
We investigated precision measurements in a two-level system coupled to a single-photon-added coherent state (SPACS) under postselected weak measurement. We analyzed strategies for improving measurement precision, including parameter estimation and the signal-to-noise ratio (SNR) in postselected weak measurements, using the photon statistics of SPACS as the meter. Our results demonstrate that SPACS-based postselected weak measurements can outperform conventional measurement schemes in terms of precision. Additionally, we explicitly introduced an alternative weak measurement method, which is commonly applied in dispersive light-atom interactions. Our work presents a novel approach to addressing fundamental issues in quantum precision measurement based on photon statistics, and it provides a method for extracting the photon and phase shifts of radiation fields through the weak values of system observables.
Weak measurement has ignited extensive research and discussions in the field of quantum metrology due to its sufficient potential in quantum measurement applications. Recent studies demonstrate that weak measurement protocols, when integrated with quantum resources, can achieve Heisenberg scaling while maintaining technical noise advantages. However, scenarios involving dephasing noise have not been addressed, as such noise induces rapid degradation of Heisenberg scaling to the standard quantum limit. Here, we incorporate phase fluctuations, a typical dephasing noise, into three Heisenberg-scaling systems. We find that postselection invariably reduces the robustness of quantum systems. Compared to coherent pointer states, squeezed coherent pointer states can exhibit superior resilience against phase fluctuations through appropriate squeezing-parameter optimization. In addition, in entanglement-assisted systems, we can optimize the postselected angle to compensate for Fisher information loss induced by the postselection process. Our findings reveal that weak measurement exhibits inherent limitations in dephasing noise metrology compared to conventional measurement.
Postselected weak measurement has aroused broad interest for its distinctive ability to amplify small physical quantities. However, the low postselection efficiency to obtain a large weak value has been a big obstacle to its application in practice since it may waste resources, and reduce the measurement precision. An improved protocol was proposed in Pang et al., Phys. Rev. Lett. 113, 030401 (2014) to make the postselected weak measurement dramatically more efficient by using entanglement. Such a protocol can increase the Fisher information of the measurement to approximately saturate the well-known Heisenberg limit. In this paper, we review the entanglement-assisted protocol of postselected weak measurement in detail, and study its robustness against technical noises. We focus on readout errors. Readout errors can greatly degrade the performance of postselected weak measurement, especially when the readout error probability is comparable to the postselection probability. We show that entanglement can significantly reduce the two main detrimental effects of readout errors: inaccuracy in the measurement result and the loss of Fisher information. We extend the protocol by introducing a majority vote scheme to postselection to further compensate for readout errors. With a proper threshold, almost no Fisher information will be lost. These results demonstrate the effectiveness of entanglement in protecting postselected weak measurement against readout errors.