Quantum averages of weak values

We re-examine the status of the weak value of a quantum mechanical observable as an objective physical concept, addressing its physical interpretation and general domain of applicability. We show that the weak value can be regarded as a definite mechanical effect on a measuring probe specifically designed to minimize the back reaction on the measured system. We then present an interesting framework for general measurement conditions (where the back reaction on the system may not be negligible) in which the measurement outcomes can still be interpreted as quantum averages of weak values. We show that in the classical limit, there is a direct correspondence between quantum averages of weak values and posterior expectation values of classical dynamical properties according to the classical inference framework.

Figure 1

Decomposition of event distribution (dark shading) for a ${\widehat{\sigma }}_z$ measurement from a PME defined by a Gaussian apparatus state with $\Delta q=\pi$ and an initial eigenstate of $\widehat{\mathit{\sigma }}\bullet {\mathit{n}}_1$ with eigenvalue $+1$, ${\mathit{n}}_1=(\sin \theta ,0,\cos \theta )$, and $\theta =11\pi ∕24$. Top, decomposition according to Eq. (2.5); bottom, actual decomposition in terms of event distributions from the two PPME’s defined by a post selection measurement of $\widehat{\sigma }\bullet {\mathit{n}}_2$, where ${\mathit{n}}_2=(-\sin \theta ,0,\cos \theta )$. The event distribution for the unlikely final outcome $\widehat{\sigma }\bullet {\mathit{n}}_2=+1$ (lightest shading) is overwhelmingly concentrated beyond the range of the spectrum of ${\sigma }_z$, indicated by dotted vertical lines.

Figure 2

Geometric construction for the weak spin vector $\mathcal{J}$ for initial and final spin-$j$ coherent states $\mid {\mathit{n}}_1;j⟩$ and $\mid {\mathit{n}}_2;j⟩$.

Figure 3

Orbit of the weak spin vector for a ${\widehat{J}}_z$ measurement, assuming the $z$ axis coincides with the direction of $\mathcal{J}\left(0\right)$.

Figure 4

Weak value ${\mathcal{J}}_z\left(q\right)$ and log-likelihood $\ln {\mathcal{P}}_{12}\left(q\right)$ for $j=20$ and ${\widehat{J}}_z$ measurement with ${\mathit{n}}_1=(0,1∕\sqrt{2},1∕\sqrt{2})$ and ${\mathit{n}}_2=(0,-1∕\sqrt{2},1∕\sqrt{2})$, as depicted in Fig. 3. Shaded areas represent regions where the weak value lies outside the bounds of the spectrum $[\mid {\mathcal{J}}_z\left(q\right)\mid >j]$.

Figure 5

Interplay between prior distribution in $q$ and the likelihood factor, and resulting consequences on the pointer distribution (see text). (a–g) on the left-hand side show prior (dotted) and posterior (solid) distributions in $q$, while figures on the right show the corresponding prior (dotted) and final posterior (solid) distributions in $p$. The top figure shows the curve of the weak value. Shaded regions correspond to regions of “eccentric” effects.

Figure 6

A weak to strong measurement transition illustrated by priors of increasing width in $q$ around $q=0$. Graphs follow the same convention as those of Fig. 5.