Strong Measurements Give a Better Direct Measurement of the Quantum Wave Function

Weak measurements have thus far been considered instrumental in the so-called direct measurement of the quantum wave function [J. S. Lundeen, Nature (London) 474, 188 (2011).]. Here we show that a direct measurement of the wave function can be obtained by using measurements of arbitrary strength. In particular, in the case of strong measurements, i.e., those in which the coupling between the system and the measuring apparatus is maximum, we compared the precision and the accuracy of the two methods, by showing that strong measurements outperform weak measurements in both for arbitrary quantum states in most cases. We also give the exact expression of the difference between the original and reconstructed wave function obtained by the weak measurement approach; this will allow one to define the range of applicability of such a method.

Figure 1

Scheme of the original DWT method used to measure the wave function.

Figure 2

Accuracy of the DWT: we show the probability $p_W$ of having ${\tilde{\psi }}_W<0$ and the probability $p_{\mathcal{D}}$ of having an error $\mathcal{D}$ larger than 0.1. The inset shows trace distance $\mathcal{D}$ in a function of ${\sigma }_{\psi }/\tilde{\psi }$ for different values of $\theta$. We randomly choose ${10}^6$ wave functions in a $d=10$-dimensional Hilbert space. Dashed lines in the inset represent the curves $D={\epsilon }_{\theta }({\sigma }_{\psi }/\tilde{\psi })$.

Figure 3

Ratio of statistical errors $(\delta {\psi }_S/\delta {\psi }_W)$ in a function of ${\tilde{\psi }}_W$. The shaded area represents the points in which the DWT is convenient with respect to the DST method, corresponding to ${\tilde{\psi }}_W\ge 0$ and $\delta {\psi }_W\le \delta {\psi }_S$.

Figure 4

Ratio of statistical errors $(\delta {\psi }_S/\delta {\psi }_W)$ in a function of $\mathcal{D}$. The shaded area represents the wave functions for which the statistical error of the DWT is lower than the DST method.

Figure 5

Mean values of $(\delta {\psi }_S/\delta {\psi }_W)$ and $\mathcal{D}$ averaged over ${10}^6$ random wave functions in a function of $\theta$.