# Measurements of Entropic Uncertainty Relations in Neutron Optics
## 3 Measurement of Entropic Noise-Noise Uncertainty Relation
### Measurement Results  
To determine the noise, all probabilities must be measured. This was done by counting the neutrons in a certain configuration each. The frequency of occurrence of an event in relation to all other possible events gives the probability. The Figures 5 and 6 show the raw form of the measured data in the case of the input state \(|+a\rangle\), respectively \(|+b\rangle\). Each bar in the chart corresponds to the counted neutrons for a particular \(m\) and either an angle \(\theta\) or a weighting \(q\). If one denotes the counted neutrons as \(I_{a,m}\) and \(I_{b,m}\) then measuring the count rate for all combinations \(a=\{-1,+1\}\), \(b=\{-1,+1\}\) and \(m=\{1,2,3,4\}\) allows to obtain all necessary probabilities via  
\[p(a,m)=\frac{I_{a,m}}{\sum_{a,m}I_{a,m}}\,\qquad p(b,m)=\frac{I_{b,m}}{\sum_{b,m}I_{b,m}} \tag{21}\]and  
\[p(a|m)=\frac{I_{a,m}}{\sum_{a}I_{a,m}}\,\qquad p(b|m)=\frac{I_{b,m}}{\sum_{b}I_{b,m}}. \tag{22}\]  
The regarded observable are \(A=\vec{a}\cdot\vec{\sigma}Y\coloneqq\sigma_{z}\) and \(B=\vec{b}\cdot\vec{\sigma}\coloneqq\sin(\beta)\sigma_{y}+\cos(\beta)\sigma_{z}\), the second operator is therefore only varied over the y-z plane. The first measurement was made in the projective case \(q=1\) so that Equation (18) becomes \(M_{m}=\{P_{+}(\vec{\tau}_{1}),P_{-}(\vec{\tau}_{1}),0,0\}\). The Bloch vector \(\vec{\tau}_{1}=(0,\sin\theta_{1},\cos\theta_{1})^{T}\) is rotated from \(0\) to \(\pi\), hence depending on how close \(\vec{\tau}_{1}\) is to \(\vec{a}\) or \(\vec{b}\) the count rate is either large or small. Figure 5 shows the neutron count rate in the MUB case for the input states Figure 5a \(|+a\rangle=|+z\rangle\) and Figure 5b \(|+b\rangle=|+y\rangle\). In order to obtain the total noise, the intensity distribution for the negative eigenstates is also necessary, which is not shown here. For each angle \(\theta\) the four outcomes are detected each for the positive and negative eigenvectors with a total of 2500 neutrons recorded in 60 seconds. Figure 5: The neutron count rate for (**a**) \(I_{+a,m}\) and (**b**) \(I_{+b,m}\) of the four-outcome POVM at a probability weighting of \(q=1\) in the case \(A=\sigma_{z}\) and \(B=\sigma_{y}\). Each histogram illustrates the change of counted neutrons depending on the angle \(\theta\) of the measurement direction \(\vec{\tau}_{1}=(0,\sin\theta_{1},\cos\theta_{1})^{T}\) with an increment of \(\Delta\theta_{1}=20^{\circ}\). The maximal height in the 60s measurement are approximately 2500. At the initial angle \(\theta_{1}=0\), outcome \(m=1\) corresponds to the projector \(P_{+}(\vec{\tau}_{1})=P_{+}(\vec{a})\) and therefore a maximal count rate occurs in (**a**), while \(m=2\) is associated with \(P_{-}(\vec{\tau}_{1})=P_{-}(\vec{a})\) and no neutrons are detected. A rotation of the projectors by angle \(\theta\) leads to a decrease of counts in \(m=1\) and an increase of measured events in \(m=2\) for the input state \(|+a\rangle\). Likewise, (**b**) shows the histogram for an input state \(|+b\rangle\), which is a distribution phase-shifted to \(I_{+a,m}\) by \(\theta=90^{\circ}\). Outcomes \(m=3\) and \(m=4\) from Equation (18) do not occur in the projective limit \(q=1\). In contrast to Figure 5 the bar chart in Figure 6 illustrates the alteration of count rates for varying statistical mixture \(q\) in Equation (18) but with static, unchanged measurement directions \(\vec{r}_{1}=\vec{e}_{z}\) and \(\vec{r}_{2}=\vec{e}_{y}\). This means that if \(q\) is 0.7 for example, the neutron spins were measured by the operators \(M_{3}\) and \(M_{4}\) with a prefactor of 0.3, hence all four outcome slots are contributing to the statistics.