# Measurements of Entropic Uncertainty Relations in Neutron Optics
## 2 Theory  
For an observable \(A\) with finite, non-degenerate spectrum \(\{a_{i}\}\) the expectation value is defined by \(\langle A\rangle=\sum_{i=1}^{N}a_{i}p(a_{i})\) where the probability \(p(a_{i})=|\langle a_{i}|\psi\rangle|^{2}\) is given by [PERSON]'s rule. The standard deviation quantifies the statistical dispersion around the mean in many repetitions of the same configuration, written out as  
\[\Delta(A)=\sqrt{\sum_{i=1}^{N}p(a_{i})(a_{i}-\langle A\rangle)^{2}}. \tag{6}\]  
There are of course other quantities that can describe the dispersion, peakedness or uniformity of probability measures. Associate to the outcomes \(a_{i}\) the random variable \(\mathbb{A}\), then the Shannon entropy [34] of the distribution is defined by  
\[H(\mathbb{A})=-\sum_{i=1}^{N}p(a_{i})\log(p(a_{i})). \tag{7}\]  
This entropy quantifies the average information content of probabilistic data and becomes larger the more uniform the distribution is. The striking difference between the standard deviation and the Shannon entropy is that the former explicitly depends on the outcome value \(a_{i}\), while the latter depends on nothing but the probabilities \(p(a_{i})\). Considering a probability distribution as a pair of values \((a_{i},p(a_{i}))\), it is quickly realized that the Shannon entropy is not affected by any relabeling or shuffling of \(a_{i}\). The entropic noise operator \(N(\mathcal{M},A)\) quantifies the average information that the apparatus \(\mathcal{M}\)[35] gains in a measurement of the operator \(A\). For that purpose an a priori distribution of eigenstates \(\{|a_{i}\rangle\}\) is considered which are prepared with uniform probability \(p(a_{i})=1/d\), where \(d\) denotes the dimension of the system state. This source of equiprobable input states is then sent to  
Figure 3: Experimental procedure to measure observables \(A\) or \(B\) in a successive manner. Instead of precisely measuring \(A\) an approximate measure of \(A\), denoted as \(O_{A}\) is performed in order to reduce the disturbance on \(B\). the instrument \(\mathcal{M}\), modeled as a completely positive, trace non-increasing quantum operation with outcomes \(m_{j}\) belonging to \(\mathbb{M}\). The noise operator is defined as the conditional entropy \(H(\mathbb{A}|\mathbb{M})\),  
\[N(\mathcal{M},A):=H(\mathbb{A}|\mathbb{M})=-\sum_{i,j}p(a_{i},m_{j})\log(p(a_{ i}|m_{j})). \tag{8}\]  
This is to be interpreted as the possibility to infer the unknown eigenstates of \(A\) by the instrument. If the outcomes \(m_{j}\) are perfectly correlated to \(a_{i}\), then a certain guess of the initial state can be made, otherwise, if the instrument can not discriminate the input states perfectly then the measurement is considered as noisy and not well calibrated. Let the observable B be an operator that generally does not commute with A. The entropic disturbance operator \(D(\mathcal{M},B)\) is modeled similar to the noise operator. Uniformly distributed eigenstates \(\{|b_{i}\rangle\}\) with eigenvalues \(b_{i}\) associated with random variable \(\mathbb{B}\) are fed to the same instrument \(\mathcal{M}\) from which a post-measurement state \(\rho_{m_{j}}=\mathcal{M}_{m_{j}}(|b_{i}\rangle\langle b_{i}|)/\text{Tr}( \mathcal{M}_{m_{j}}(|b_{i}\rangle\langle b_{i}|))\) emerges. In the disturbance configuration there is, however, an additional subsequent measurement of observable \(B\) with outcomes \(\{b^{\prime}_{j}\}\). Due to the disturbing nature of the measurement apparatus \(\mathcal{M}\), generally, a loss of correlation occurs. A subtle, yet important addendum to the concept of disturbance are error corrections. After measurement by \(\mathcal{M}_{m_{j}}\), the state decomposed to the eigenstates of the measurement observables can be further transformed by a quantum operation \(\mathcal{E}_{m_{j}}\) dependent on the pointer value \(m_{j}\) of the apparatus.