Full counting statistics of weak-value measurement

A weak measurement consists of coupling a system to a probe in such a way that constructive interference generates a large output. So far, only the average output of the probe and its variance were studied. Here, the characteristic function for the moments of the output is provided. The outputs considered are not limited to the eigenstates of the pointer or of its conjugate variable, so the results apply to any observable $\widehat{o}$ of the probe. Furthermore, a family of well-behaved complex quantities, the normal weak values, is introduced, in terms of which the statistics of the weak measurement can be described. It is shown that, within a good approximation, the whole statistics of weak measurement is described by a complex parameter, the weak value, and a real one.

Figure 1

A schematic view of the measurement with pre- and postselection, the horizontal direction representing increasing time. A system is prepared in ${\rho }_i$, interacts with a probe, and then is projected into $|S\rangle$ by a measurement. After the interaction, the observable $o$ of the probe is measured. A random number generator produces a number $0\le x\le 1$. If $x>w\left(S\right)$, an arbitrarily chosen probability depending on $S$, then the record $o$ is discarded.

Figure 2

Comparison between the interpolating formula Eq. (39), the simpler Eq. (40), and the exact result for the output $p$ of a probe weakly measuring a spin component $\mathbf{n}·\sigma$, as a function of the angle between the preselected state $|{\mathbf{n}}_i:\uparrow \rangle$ and the postselected one $|{\mathbf{n}}_f:\uparrow \rangle$, ${\mathbf{n}}_f·{\mathbf{n}}_i=\cos \theta$ for a fixed preselection, with $\mathbf{n}·{\mathbf{n}}_i=0$ and all vectors coplanar. The panels correspond to different interaction strengths: (a) $\lambda /\delta p=0.5$, (b) $\lambda /\delta p=0.1$.

Figure 3

Comparison between the interpolating formula (solid symbols) Eq. (22), and the exact result (open symbols) Eq. (49) for the output ${\langle p^n\rangle }_f-\overline{p^n}$ in units of $\overline{p^n}$ as a function of the angle between the preselected state $|{\mathbf{n}}_i:\uparrow \rangle$ and the postselected one $|{\mathbf{n}}_f:\uparrow \rangle$, ${\mathbf{n}}_f·{\mathbf{n}}_i=\cos \theta$ for a fixed preselection, with $\mathbf{n}·{\mathbf{n}}_i=0$ and all vectors coplanar. Interaction strength: $\lambda /\delta p=\lambda /\Delta P=0.1$. For the values of $\theta$ not shown the discrepancy is undiscernible to the naked eye.

Figure 4

Universal scaling for the moments.