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

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.

Figure 1

A sketch of the distribution functions when $\sigma =1$ and $g=1.5$. The blue (left) curve is the probe distribution before the interaction. The red (upper right) and the yellow (lower right) curves are the ones after the interaction with postselection and without postselection, respectively. We choose the initial state of the measured system as $|i\rangle =\left(\right|+\rangle +|-\rangle )/\sqrt{2}$, and the postselected state which satisfies the weak value $\mathrm{Re}{\langle \widehat{A}\rangle }_w=0$ and $|{\langle \widehat{A}\rangle }_w|=5$.

Figure 2

The contour plots (a), (b) for $\mathrm{Pr}\left[{\mathcal{E}}_{2,\mathrm{ps}}\right]/\mathrm{Pr}\left[{\mathcal{E}}_{2,\mathrm{nps}}\right]$ and (c), (d) for ${\beta }_{\mathrm{ps}}\left(g\right)/{\beta }_{\mathrm{nps}}\left(g\right)$ in which the horizontal axis indicates the absolute value of the weak value $\left|{\langle \widehat{A}\rangle }_w\right|$. The left- and right-side graphs have the vertical axis indicating the coupling constant divided by the initial fluctuation $g/\sigma$ and the critical point $c$, respectively. In (a) and (b), the darker blue indicates the smaller value. In (c) and (d), the darker red indicates the larger value.

Figure 3

Plots of the wave functions $f_{\mathrm{ps}}\left(x\right|g)$ (the red dashed curve) and $f_{\mathrm{nps}}\left(x\right|g)$ (the orange solid curve) against $x$ in three coupling constant cases: (a) $g=1$, (b) $g=2$, and (c) $g=5$. In the case (c), two plots are almost overlapped. In these graphs, the other parameters are fixed as follows: $\mathrm{Re}{\langle \widehat{A}\rangle }_w=0,\left|{\langle \widehat{A}\rangle }_w\right|=5,{\left|\langle +|i\rangle \right|}^2={\left|\langle -|i\rangle \right|}^2=1/2$, and $\sigma =1$.