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弱値増幅における最適プローブと統計的検定

友紀 須佐, Yuki Susa

Published Mar 1, 2016• 0 citations
OpenAlex

Abstract

Recently, the weak-value amplification (WVA) has been extensively investigated as a promising quantum technique for improving an accuracy of a precision measurement.The basic idea of WVA comes from the weak measurement introduced by Aharonov, Albert, and Vaidman in 1988.The weak measurement is the indirect quantum measurement with the weak interaction and postselection of the final state of the measured system.The attention point is that we can measure the quantity called the weak value, which appears in the shift of the measuring probe given by the weak measurement.Because the weak value can be outside the range of the eigenvalues, the shift of the measuring probe obtained by the weak measurement becomes larger than the one given by the conventional measurement.This shift amplification effect by the weak value is the WVA, which has been studied for precisely measuring the small coupling constant.In this thesis, we propose the two possible applications of the WVA.First, we analytically obtain the optimal probe wave function for a given weak value by the variational method.It is shown that the amplification factor obtained by the optimal probe has noupper bound while the one in the Gaussian probe already reported by some researchers has the upper bound.Furthermore, the variance of the optimal probe after the measurement can be zero.We can derive the optimal probe by the Lagrange multiplier method for maximizing the shift of the probe position or minimizing the variance of the final distribution.Second we have developed the way which has a technical advantage of the WVA from the viewpoint of statistics.It is often argued that the WVA is not helpfull for the parameter estimation.Therefore we consider the statistical inferences that the WVA will be advantageous.We propose a method to determine whether the interaction is present or not and evaluate the capability of the WVA for this interaction detection problem by using the hypothesis testing in statistics.As the main result, it is shown that the merit of the WVA is the reduction of the possibility to miss the presence of the interaction more than the conventional measurement while keeping the probability of a misdetection, when the absolute value of the weak value is outside the range of the eigenvalues.In this discussion, we assume that the number of the obtained data is infinitely large and we neglect the data loss by the failure of the postseleciton.iii the deepest appreciation to my ex-supervisor

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Recently, the weak-value amplification (WVA) has been extensively investigated as a promising quantum technique for improving an accuracy of a precision measurement.The basic idea of WVA comes from the weak measurement introduced by Aharonov, Albert, and Vaidman in 1988.The weak measurement is the indirect quantum measurement with the weak interaction and postselection of the final state of the measured system.The attention point is that we can measure the quantity called the weak value, which appears in the shift of the measuring probe given by the weak measurement.Because the weak value can be outside the range of the eigenvalues, the shift of the measuring probe obtained by the weak measurement becomes larger than the one given by the conventional measurement.This shift amplification effect by the weak value is the WVA, which has been studied for precisely measuring the small coupling constant.In this thesis, we propose the two possible applications of the WVA.First, we analytically obtain the optimal probe wave function for a given weak value by the variational method.It is shown that the amplification factor obtained by the optimal probe has noupper bound while the one in the Gaussian probe already reported by some researchers has the upper bound.Furthermore, the variance of the optimal probe after the measurement can be zero.We can derive the optimal probe by the Lagrange multiplier method for maximizing the shift of the probe position or minimizing the variance of the final distribution.Second we have developed the way which has a technical advantage of the WVA from the viewpoint of statistics.It is often argued that the WVA is not helpfull for the parameter estimation.Therefore we consider the statistical inferences that the WVA will be advantageous.We propose a method to determine whether the interaction is present or not and evaluate the capability of the WVA for this interaction detection problem by using the hypothesis testing in statistics.As the main result, it is shown that the merit of the WVA is the reduction of the possibility to miss the presence of the interaction more than the conventional measurement while keeping the probability of a misdetection, when the absolute value of the weak value is outside the range of the eigenvalues.In this discussion, we assume that the number of the obtained data is infinitely large and we neglect the data loss by the failure of the postseleciton.iii the deepest appreciation to my ex-supervisor
The novel notion of a physical quantity called 'weak value', which was proposed by Aharonov et al. in 1988, has attracted much attention in the fields of quantum foundations and quantum measurement.In particular, the fact that through proper state selections the weak value can be made arbitrarily large, exceeding the conventional bound set by the maximum eigenvalue of the physical observable, has been widely regarded as a promising source for precision measurement.Indeed, in recent years we have seen increasingly many reports of successful applications of the weak value in photonic systems with various instances of observables amplified effectively.However, it has also been recognized that there are two major obstacles in realizing this weak value amplification (WVA).First, it is unclear under what conditions the WVA becomes effective due to the conflict between the ambiguity caused by the state selections and the amplification effect.Second, the required state selections may not be available depending on the systems one is dealing with.In order to expand the applicability of the WVA, one must solve these problems properly.In this dissertation, we first review the basics of precision measurement and the WVA and then present our method of evaluation for the WVA considering the measurement uncertainty including the ambiguity of the WVA.We applied our method of evaluation for the two celebrated experiments to confirm the basic validity of the results of the two experiments.In addition, as a new application of the WVA, we consider the time evolution of unstable particles to find that the decay time distribution is generally characterized by the weak value of the Hamiltonian and that the state selections can indeed be performed by measuring the particles produced during the decay.Our result indicates that the WVA may also be applicable to the system of unstable particles.
The weak measurement proposed by Aharonov and his colleagues extracts information about a physical quantity of the system by the postselection as shifts of the argument of the probe wave function. The more the postselected state is orthogonal to the preselected state, the larger the shift determined by the weak value becomes. Recently, the signal amplification by the weak measurement has been extensively studied. In the present work, we explicitly obtain the optimal probe wave function and the amplification factor for a given weak value, which is calculated from the experimental setup. It is shown that the amplification factor has no upper bound, in contrast to the Gaussian probe wave function, and that the amplified signal is sharp.
We show using statistically rigorous arguments that the technique of weak value amplification (WVA) does not perform better than standard statistical techniques for the tasks of single parameter estimation and signal detection. Specifically we prove that post-selection, a necessary ingredient for WVA, decreases estimation accuracy and, moreover, arranging for anomalously large weak values is a suboptimal strategy. In doing so, we explicitly provide the optimal estimator, which in turn allows us to identify the optimal experimental arrangement to be the one in which all outcomes have equal weak values (all as small as possible) and the initial state of the meter is the maximal eigenvalue of the square of the system observable. Finally, we give precise quantitative conditions for when weak measurement (measurements without post-selection or anomalously large weak values) can mitigate the effect of uncharacterized technical noise in estimation.
The weak measurement proposed by Aharonov and his colleagues extracts information of a physical quantity of the system by the post selection as the shifts of the argument of the probe wavefunction. The shift is called the weak value and is larger for the post-selected state more orthogonal to the initial state. Recently, the signal amplification by the weak measurement has been extensively studied. In the present work, we explicitly obtain the optimal probe wavefunction and the amplification factor for a given weak value, which is calculated from the experimental setup. It is shown that the amplification factor has no upper bound in contrast to the Gaussian probe wavefunction and that the amplified signal is sharp.
Weak value amplification (WVA) is a measurement technique in which the effect of a pre- and post-selected system on a weakly interacting probe is magnified. In this thesis, I present the first experimental observation of WVA of a single photon. We observed that a signal photon - sent through a polarization interferometer and post-selected by photodetection in the almost-dark port - can act like eight photons. The effect of this single photon is measured as a nonlinear phase shift on a separate laser beam. The interaction between the two is mediated by a sample of laser- cooled 85Rb atoms. Electromagnetically induced transparency (EIT) is used to enhance the nonlinearity and overcome resonant absorption. I believe this work to be the first demonstration of WVA where a deterministic interaction is used to entangle two distinct optical systems. In WVA, the amplification is contingent on discarding a large portion of the original data set. While amplification increases measurement sensitivity, discarding data worsens it. Questioning whether these competing effects conspire to improve or diminish measurement accuracy has resulted recently in controversy. I address this question by calculating the maximum amount of information achievable with the WVA technique. By comparing this information to that achievable by the standard technique, where no post-selection is employed, I show that the WVA technique can be advantageous under a certain class of noise models. Finally, I propose a way to optimally apply the WVA technique.
We study the detection capability of the weak-value amplification on the basis of the statistical hypothesis testing. We propose a reasonable testing method in the physical and statistical senses to find that the weak measurement with the large weak value has the advantage to increase the detection power and to reduce the possibility of missing the presence of interaction. We enhance the physical understanding of the weak value and mathematically establish the significance of the weak-value amplification. Our present work overcomes the critical dilemma of the weak-value amplification that the larger the amplification is, the smaller the number of data becomes, because the statistical hypothesis testing works even for a small number of data. This is contrasted with the parameter estimation by the weak-value amplification in the literature which requires a large number of data.