# Measurements of Entropic Uncertainty Relations in Neutron Optics
## 4 Measurement of Entropic Noise-Disturbance Uncertainty Relation
### Measurement Results
However, as the green curve in Figure 12 implies, the situation can be improved by  
Figure 11: Measurement results of the conditional probabilities for (a) \(p(m|a)\), (b) \(p(b^{\prime}|b)\) and (c) \(p_{corr}(b^{\prime}|b)\). From \(\theta=0\) to \(\theta=\pi/2\) the increment is \(\pi/18\) and changes to \(\pi/9\) from \(5\pi/9\) on. At the angle \(\theta=0\) the measurement operator is \(\sigma_{z}\) and the probability to receive \(m=\pm 1\) given the value of the initial eigenvector is \(a=\pm 1\) is maximal in (**a**), while the results with opposite signs \(a=\mp 1\) are impossible to occur. Rotating the measurement direction leads to a sinusoidal change of the probabilities. The center (**b**) and right graph (**c**) show the probabilities for receiving \(b^{\prime}\) given state \(b\). The maxima and minima are shifted in phase to the first panel by \(\pi/2\). The uncorrected curve in the middle has not the same wide arc form compared to the curve where the optimal correction procedure is conducted. correlating the random variable \(\mathbb{B}^{\prime}\) of the second apparatus with the random output variable \(\mathbb{M}\) of the first measurement instrument. In terms of complete positive trace-preserving maps, this can be written as the operation  
\[\mathcal{E}(|m\rangle):=\left\{ \begin{array}{cc}|+b\rangle&\vec{b}\cdot\vec{r} \geq 0\\ |-b\rangle&\vec{b}\cdot\vec{r}<0\end{array} \right.. \tag{26}\]  
The green-colored result, therefore, shows the minimal disturbance, where the second apparatus makes a measurement on the state \(|\pm b\rangle\), the state rotated onto the closest eigenvector of \(B\). Finally, it remains to have a look at the plot of noise vs. disturbance. In Figure 13 three curves are shown. The red curve corresponds to the uncorrected uncertainty relation and the green one to the corrected in accordance with the color-coding of Figure 12. The white-hatched area is the region not reachable by sharp, projective, joint measurements M of the two non-commuting observables \(A,B\). The green boundary of the uncertainty region Equation (19) is saturated for this particular case, while the Maassen-Uffink like relation \(N(\mathcal{M},A)+D(\mathcal{M},B)=1\), given by the black line, displays an underestimation of the obtainable noise-disturbance values. Figure 12: Entropic noise \(N(\mathcal{M},A)\) in blue and uncorrected/optimally corrected disturbance \(D_{\mathcal{E}}(\mathcal{M},B)\) in green/red as a function of the angle \(\theta\) of the measurement operator. The noise for observable \(A=\sigma_{z}\) at \(\theta=0\) is minimal, while the eigenstates of \(|\pm b\rangle\) are maximally disturbed at this angle. Rotating the polar angle closer to the observable \(B=\sigma_{y}\) reduces the disturbance at the expanse of increasing the noise. After the first apparatus an additional rotation of the output state \(|m\rangle\) onto the closest eigenvector of \(B\) reduces the loss of correlation in parts and reduces the disturbance furthermore. At \(\theta=\pi/2\) the measurement apparatus makes a perfect measurement of \(B\) inducing no disturbance, being however maximally noisy with respect to a measurement of \(|\pm a\rangle\). However, the inequality Equation (19) is to be treated with caution, as it is indeed possible to violate it when loosening some of the restrictions made in the presented measurements. Indeed, [PERSON] et al. [32] have proven that for POVMs the noise-disturbance uncertainty relation for two-dimensional quantum systems can be further improved and \(g(N(\mathcal{M},\sigma_{i}))^{2}+g(D(\mathcal{M},\sigma_{j}))^{2}>1\) is possible. A generally tight, entropic noise-disturbance uncertainty relation for arbitrary pairs of non-commuting observables in two dimensions has however not been found or tested so far.