# Measurements of Entropic Uncertainty Relations in Neutron Optics
## 3 Measurement of Entropic Noise-Noise Uncertainty Relation
### Measurement Results
Putting the results together using Equation (8) for both observables A and B, the noise is plotted as a function of the rotation angle in Figure 7. The noises vary between 0 and 1; for \(\theta_{1}=0\) a perfect measurement of \(A\) is made by \(\mathcal{M}\) which results in a noisefree determination of the eigenstates \(|\pm a\rangle\) given by the blue data points, while on the other hand \(|\pm b\rangle\) are maximally unpredictable at this point. When the projectors \(P_{\pm}(\vec{r}_{1})=\frac{1}{2}(\mathbb{1}+\vec{r}_{1}(\theta_{1},\phi_{1}= \frac{\pi}{2})\cdot\vec{\sigma})\) are rotated closer towards observable \(B\), a trade-off between the noises occurs making the discrimination of \(|\pm a\rangle\) more difficult and the differentiation of \(|\pm b\rangle\) easier. This impossibility of jointly distinguishing the two eigenstates of the non-commuting observables of \(A,B\) is known as the measurement uncertainty principle. Generally, we were interested in the region \(R_{\mathrm{NN}}(A,B)\) Equation (13) of all obtainable pairs of noise values. For that purpose, an analytic approach to finding the minimal and maximal values was necessary. The idea behind maximizing the noise is simple: since \(\vec{a},\vec{b}\in\mathbb{R}^{3}\) are confined to the y-z plane, rotating the measurement operator \(\vec{r}_{1}^{\prime}(\theta_{1},\phi_{1})\) such that it lies outside the plane \(\vec{r}_{1}^{\prime}\cdot(\vec{a}\times\vec{b})\
eq 0\) increases the entropy. The purple points at \(\frac{\pi}{2}\) and \(\frac{\pi}{2}\) in Figure 7 are lifted to the maximal value through this procedure. To understand the strategy of finding the minimal value, it is necessary to look at the black curve representing the sum \(N(\mathcal{M},A)+N(\mathcal{M},B)\). Apparently, the sum of the entropic noises has two local minima in Figure 7 at \(\theta_{1}\simeq 5^{\circ}\) and \(\theta_{2}\simeq 74^{\circ}\) in the region 0-\(\pi/2\), calculable from the first derivative. The idea to obtain the lowest bound of the uncertainty region is to build the convex combination of the projectors \(P_{\pm}(\vec{r}_{1})\) and \(P_{\pm}(\vec{r}_{2})\) at exactly these minima. The linear progression of the noises as a function of the scalar parameter \(q\) is shown in Figure 8. The statistical weighting varies by about 0.1 and the two noises run with opposite proportionality, the sum always being at the minimum value of 0.96 (the same minima as in Figure 7). Uncertainty relations are best viewed through a parametric representation as exhibited in Figure 9. The figure shows for Bloch vector \(\vec{a}=\vec{e}_{z}\) and varying \(\vec{b}\) how the uncertainty region gradually changes. The white hatched part marks all the points which are not reachable by a joint measurement. Starting from the MUB case given in the first picture, it is seen that the amount of achievable points is sinking  
Figure 8: Theoretical lines plus measurement points of \(N(\mathcal{M},A)\) in blue and \(N(\mathcal{M},B)\) in red for \(A=\sigma_{z}\) and \(B\simeq\sin(79^{\circ})\sigma_{y}+\cos(79^{\circ})\sigma_{z}\) for varying statistical weighting \(q\). To obtain this plot the operators \(P_{\pm}(\vec{r}_{1})\) and \(P_{\pm}(\vec{r}_{2})\) are fixed to the extreme values \(\theta_{1}=5^{\circ}\) and \(\theta_{2}=74^{\circ}\) corresponding to the minima in Figure 7. The black line shows that the sum of the entropy remains constant at the lowest value of Figure 7. As a result, realizing the POVM Equation (18) reduces the overall noise. Figure 7: The measurement results of noises \(N(\mathcal{M},A)\) in blue and \(N(\mathcal{M},B)\) in red for \(A=\sigma_{z}\) and \(B\simeq\sin(79^{\circ})\sigma_{y}+\cos(79^{\circ})\sigma_{z}\) together with the theoretical predictions given by the solid curves in the projective case.