Statistical hypothesis testing by weak-value amplification: Proposal and evaluation

Abstract

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.

Physical description of statistical hypothesis testing for a weak-value-amplification experiment using a birefringent crystal

“…For the testing problem to determine whether the crystal is birefringent or not, we take the following hypotheses: the null hypothesis H 0 , in which the crystal is not birefringent (i.e., λ H = λ V ), and the alternative hypothesis H 1 , in which the crystal is birefringent (i.e., λ H = λ V ). We note that by the interaction, the refraction occurs for both hypotheses, which somewhat differs from the previous work [17].…”

“…These experimental results clearly show the physical intuition of the testing and the advantage of WVA by comparing the powers as presented in Ref. [17]. We conclude that the angle of the polarizers that give the weak value is the only factor in determining the case that the weak measurement is superior.…”

“…We also remark that condition (17) is related to the requirement for a weak value derived in Ref. [17] as | Â t w | 2 ≥ |λ − | 2 ⇔ C(α, β) ≤ 0. The experimental results are summarized in Fig.…”

“…As an application we consider the statistical hypothesis testing proposed in Ref. [17] in the experiment that uses the birefringent crystal as explained in the previous section. For the testing problem to determine whether the crystal is birefringent or not, we take the following hypotheses: the null hypothesis H 0 , in which the crystal is not birefringent (i.e., λ H = λ V ), and the alternative hypothesis H 1 , in which the crystal is birefringent (i.e., λ H = λ V ).…”

“…Reference [17] shows that WVA can be more significant for interaction detection than the measurement without postselection when the weak value is outside * susa@th.phys.titech.ac.jp the eigenvalue range of the measured system observable. The WVA is more likely than the ordinary measurement to correctly indicate when the interaction indeed exists.…”