# Measurements of Entropic Uncertainty Relations in Neutron Optics
## 2 Theory
The disturbance operator is defined as the conditional entropy \(H(\mathbb{B}|\mathbb{B}^{\prime})\),  
\[D_{\mathcal{E}}(\mathcal{M},B):=H(\mathbb{B}|\mathbb{B}^{\prime})=-\sum_{i,j}p (b_{i},b^{\prime}_{j})\log(p(b_{i}|b^{\prime}_{j})). \tag{9}\]  
The error correction is hidden in the probabilities  
\[p(b_{i},b^{\prime}_{j})=\sum_{k}p(b^{\prime}_{j}|b_{i},m_{k})p(b_{i},m_{k})= \sum_{k}\left\langle b^{\prime}_{j}|\mathcal{E}_{m_{k}}(\rho_{m_{k}})|b^{ \prime}_{j}\right.p(b_{i},m_{k}). \tag{10}\]  
The entropic disturbance is to be interpreted as the possibility to infer the unknown eigenvalues \(b_{j}\) of \(B\) from the outcomes of \(b^{\prime}_{j}\) after the intermediate apparatus \(\mathcal{M}\) induced a disturbance on the initial state and an appropriate correction transformation \(\mathcal{E}_{m_{j}}\) is made. If a perfect guess is possible from the measurement results there is no disturbance, else if the instrument has reduced the correlation to some extent the inference of the initially unknown eigenstates \(|b_{i}\rangle\) is faulty. The optimal error correction is defined as the transformation \(\mathcal{E}_{m_{j}}\) that, for a given measurement \(\mathcal{M}_{m_{j}}\), minimizes the disturbance \(D(\mathcal{M},B)=\inf_{\mathcal{E}}D_{\mathcal{E}}(\mathcal{M},B)\). From the definitions of the entropic noise and disturbance it becomes apparent that if \([A,B]\
eq 0\), it is not possible to make noiseless and disturbance-free joint measurements. Generally, two different kinds of uncertainty relations arise in this framework, first derived in [30]. * Noise-noise uncertainty relation \[N(\mathcal{M},A)+N(\mathcal{M},B)\geq-\log(\max_{i,j}|\left\langle a_{i}|b_{ j}\right\rangle|^{2}).\] (11) This tradeoff expresses the inability for the device \(\mathcal{M}\) to jointly discriminate the eigenstates \(|a_{i}\rangle\) and \(|b_{i}\rangle\) to arbitrary precision. * Noise-disturbance uncertainty relation \[N(\mathcal{M},A)+D_{\mathcal{E}}(\mathcal{M},B)\geq-\log(\max_{i,j}|\left\langle a _{i}|b_{j}\right\rangle|^{2}).\] (12)This inequality implies that the measurement apparatus cannot be noise and disturbance-free for both eigenstates of the non-commuting operators \(A\) and \(B\). At this point, it makes sense to point out a difference in the uncertainty relations involving entropy or standard deviations, respectively. From Equations (6) it is apparent that the formula depends generally on the input state \(|\psi\rangle\), while the definitions of noise and disturbance involve only the observables' eigenstates which makes Equation (11) and (12) state independent. Furthermore, the noise-noise and noise-disturbance inequalities look identical to the well-known Maassen-Uffink [15] relation which describes preparation uncertainties. One of the biggest challenges in exploring uncertainty relations is finding the smallest boundary and the saturation conditions, i.e., the conditions under which the uncertainty relations acquire equality. Optimization studies are rather tough and therefore tight solutions have been proven only for a few selected cases. The Maassen-Uffink relation, for example, is only saturated for mutually unbiased (MUB) states and can not reach equality for other states. In the approach [32] the region of obtainable values is written out as  
\[R_{NN}(A,B) =\{N(\mathcal{M},A),N(\mathcal{M},B)\}\, \tag{13}\] \[R_{ND}(A,B) =\{N(\mathcal{M},A),D_{\mathcal{E}}(\mathcal{M},B)\}. \tag{14}\]  
Finding this set of points reachable by joint measurements is an ongoing task. For two-dimensional quantum systems promising results have been derived in the first instance.