# Measurements of Entropic Uncertainty Relations in Neutron Optics
## 4 Measurement of Entropic Noise-Disturbance Uncertainty Relation
### Measurement Results  
The evaluation of the data is similar to the methods of the last section. However, each count rate has now three subscripts \(I_{a,m,b^{\prime}}\), as well as \(I_{b,m,b^{\prime}}\), the first index denoting the prepared eigenstate and the latter two indicate the result of the outcomes of the \(m\) and \(b^{\prime}\)-measurements. The conditional and marginal probabilities required for the entropies are  
\[p(a)=\frac{\sum_{m,b^{\prime}}\,I_{a,m,b^{\prime}}}{\sum_{a,m,b^{ \prime}}\,I_{a,m,b^{\prime}}}\,\qquad\quad p(b)=\frac{\sum_{m,b^{\prime}}\,I_{b,m,b^{ \prime}}}{\sum_{b,m,b^{\prime}}\,I_{b,m,b^{\prime}}}\, \tag{23}\] \[p(m|a)=\frac{\sum_{b^{\prime}}\,I_{a,m,b^{\prime}}}{\sum_{m,b^{ \prime}}\,I_{a,m,b^{\prime}}}\,\qquad p(b^{\prime}|b)=\frac{\sum_{m}\,I_{b,m,b^{ \prime}}}{\sum_{m,b^{\prime}}\,I_{b,m,b^{\prime}}}. \tag{24}\]  
Figure 10: Experimental realization for the measurement of noise \(N(\mathcal{M},A)\) and disturbance \(D_{\mathcal{E}}(\mathcal{M},B)\). In the preparation phase, the monochromatic neutrons are transformed to the observables’ eigenstates, indicated by the left Bloch sphere. DC-2 controls the incident angle \(\theta\) of the spin. As noted in the middle Bloch sphere, the analyzer transmits best in the z-direction and gets worse as the operator \(M=\sin\theta\,\varphi_{y}+\cos\theta\,\sigma_{z}\) turns away from the poles. DC-3 can either be used to produce the uncorrected eigenstates \(|\pm m\rangle\) or perform the optimal correction and generate the spins \(|\pm b\rangle=|\pm y\rangle\). This different mode of operation is characterized by the light green color mark. In the dark green area, the second, subsequent projection takes place, in which DC-4 transforms the incoming vectors in such a way that \(B\) is measured optimally. Unlike for noise-noise uncertainty relations, an error correction procedure of the post measurement state \(|m\rangle\) must be taken into account which can drastically change the degree of disturbance. Setting the experimental observable to \(A=\sigma_{z}\), \(B=\sigma_{y}\) and \(M=\cos\theta\,\sigma_{y}+\sin\theta\,\sigma_{z}\) the oscillation of the conditional probabilities depending on the angle \(\theta\) have been recorded and are plotted in Figure 11. The left panel Figure 11a illustrates a sine/cosine behavior as expected for the probability of preparing the input state corresponding to the eigenvalue \(a\) and receiving measurement outcome \(m\). Based on the course of the conditional probabilities, one realizes for \(0\leq\theta\leq\pi\) that \(|p(b^{\prime}|b)-\frac{1}{2}|\leq|p_{opt}(b^{\prime}|b)-\frac{1}{2}|\), that is the modulus of the probability (centered at \(\frac{1}{2}\)) when leaving the post measurement state as \(|m\rangle\) is surpassed by the curve, where the optimal error correction is employed. The comparison of the respective curves and the shape of the noise can be seen well in Figure 12. The theoretical prediction of noise and uncorrected/optimally corrected disturbance for projective measurements are  
\[N(\mathcal{M},A)=h(\cos\theta)\,\qquad D_{uncorr}(\mathcal{M},B)=h(\sin^{2} \theta)\,\qquad D_{opt}(\mathcal{M},B)=h(\sin\theta). \tag{25}\]  
The interpretation of the blue noise curve is all clear by now: the measurement starts perfectly correlated leading to vanishing noise and loses its correlation the closer \(M\) is to observable \(B\) before further rotation increases the anti-correlation and makes \(N(\mathcal{M},A)\) smaller again. The disturbance has a reciprocal form to the noise. The information about the initial states is maximally destroyed by the first apparatus and there is no certainty about the second measurement outcome. The more approximate \(B\) is measured in the first apparatus the better the correlations are conserved. The red-colored data points show \(D_{uncorr}(\mathcal{M},B)\), the disturbance when no correction is made and the \(b^{\prime}\)-measurement is made on \(|m\rangle\).