Inaccurate (weak) measurements classical and quantum

Abstract

We consider highly inaccurate measurements made on classical stochastic and quantum systems. In the quantum case such a \e{weak} measurement preserves coherence between the system's alternatives. We demonstrate that in both cases the information about the scenario realised in each individual trial is lost. However, ensemble parameters such as classical path probabilities, and quantum quasi-probabilities can be extracted from the obtained statistics. In both cases causality ensures that additional post-selection only redistributes individual outcomes between the system's final states. Quantum quasi-probabilities may change sign, which allows for anomalously large meter's (pointer's) reading for some final states. These, we show, result from mere \e{reshaping} of a broad distribution obtained earlier, and provide no \e{experimental evidence} of quantum variables taking, on rare occasions, exceptionally large values.

Figures (5)

• Figure 1: Two broad Gaussians (\ref{['5']}) in the l.h.s. of Eq.(\ref{['1']}), $\Delta x=30$, $A_1=1$, $B_1=0$, and $A_2=-0.8$, $B_2=-1$, add up to a smaller Gaussian shifted to the right by approximately 4 units. The dot-dashed line shows the same Gaussian centred at $x=0$. In the inset the added dashed line shows, for comparison, the Gaussian in the r.h.s. of Eq.(\ref{['1']}).
• Figure 2: a) Alice's classical setup ($N=3$). The particle passes through a state $|i {\rangle}$ on the way to its final state $j$. b) Each of the nine paths available to the particle is equipped with the the probability in Eq.(\ref{['1a']}). c) initial uncertainty of an accurate pointer's position, $\Delta x$, is small compared to the differences $max |B_i|$. d) for an inaccurate pointer one has $\Delta x >max |B_i|$.
• Figure 3: a) Alice's quantum setup ($N=3$). A system passes through a state $|b_i {\rangle}$ on the way to its final state $|f_j{\rangle}$. b) Each of the nine virtual paths available to the system is equipped with with the amplitude in Eq.(\ref{['1c']}). Quasi-probabilities (\ref{['1e']}) can also be ascribed without destroying coherence between the paths (see Sect.VI)
• Figure 4: The direction along which the intermediate states $|b_{1,2}{\rangle} =|n_{\uparrow,\downarrow}{\rangle}$ are quantised is parametrised by its azimuthal and polar angles, $\phi$ and $\theta$. The system $(N=2)$ is prepared in $|z_\uparrow{\rangle}$ and post-selected in $|n'_\uparrow{\rangle}$ or $|n'_\downarrow{\rangle}$, where $\phi'=0$ and $\theta'=0.95\pi$. Inside the yellow region all four path "probabilities" in Eqs.(\ref{['f1']}) are positive. Elsewhere, at least one of them has a negative value.
• Figure 5: a) Distributions of the readings of a highly inaccurate pointer, $\Delta x_1/(B_2-B1)=50$ measuring spin's projection onto the direction $\phi=\pi$, $\theta=\pi/2$ . The larger filled curve, centred at $x_1=0$, is the distribution without post-selection (\ref{['6c']}), with the dot-dash line showing the approximation (\ref{['4a']}). The smaller filled curve, centred at $x_1=-12.6051$ [cf. Eq.(\ref{['h2']})], is the unnormalised distribution [cf. Eq.(\ref{['2d']})] conditioned on post-selection in the state $|n'_\uparrow{\rangle}$ polarised along the direction $\phi'=0$, $\theta'=0.95\pi$. The dashed line is the approximation (\ref{['3d']}).