# Measurements of Entropic Uncertainty Relations in Neutron Optics
## 1 Introduction
The standard deviations are then defined as \((\Delta q)^{2}=\langle(q-\langle q\rangle)^{2}\rangle\) and \((\Delta p)^{2}=\langle(p-\langle p\rangle)^{2}\rangle\), where \(\langle q\rangle=\int q|\psi(q)|^{2}\mathrm{d}q\) and \(\langle p\rangle=\int p|\phi(p)|^{2}\mathrm{d}p\), which gives \(\Delta q=a/\sqrt{3}\) and \(\Delta p=\infty\). So if we take these as measures of uncertainty, we have to say that the momentum is completely uncertain (independent of \(a\)). The problem stems from the definition of \(\Delta p\) where the probability density \(|\phi(p)|^{2}\) is multiplied by a factor \(p^{2}\), giving increasing weight to the distant parts of the probability distribution. Although the tails of the momentum distribution contain very little probability, they fall off too slowly for \(\Delta p\) to  
Figure 1: Experimental procedure to measure preparation uncertainty relations. Each particle that is emitted by the source preparing identical states \(|\psi\rangle\) either undergoes a measurement of observable \(A\) or \(B\). be finite, which illustrates that the standard deviation is not a proper measure of the width of a general probability distribution (even if its good for Gaussians). The uncertainty relation as formulated by [PERSON] in terms of standard deviations \(\Delta(A,|\psi\rangle)\Delta(B,|\psi\rangle)\geq\frac{1}{2}|\langle\psi|[A,B]|\psi\rangle|\) has two flaws: (i) the standard deviation is not an optimal measure for all states, as discussed above. (ii) the boundary (right-hand side of any uncertainty relation) can become zero for non-commuting observables (this is also the case for our neutron spins for a combination of \(|\psi\rangle=|x\rangle\), \(A=\sigma_{x}\) and \(B=\sigma_{y}\)). At this point, it should be mentioned, that the reason for the existence of that bound is the well known canonical commutation relations (CCR) of position and momentum of a quantum particle. An algebraic structure in the form \(\mathcal{U}(A,B,\psi)\geq\mathcal{B}(A,B)\) (in words: the uncertainty \(\mathcal{U}\) of the observables for a quantum state, must always be larger than some state-independent bound \(\mathcal{B}\)) is desirable for an uncertainty relation. Note that [PERSON]'s (and [PERSON]'s) inequality \(\Delta Q\Delta P\geq\frac{\hbar}{2}\) has that form, but its generalization Equation (2) does not. In order to represent a quantitative physical notion of uncertainty, \(\mathcal{U}\) must at least possess the following elementary property: If and only if \(\psi\) is a simultaneous eigenstate of \(A\) and \(B\) may \(\mathcal{U}\) become zero. From this, we can infer a property of \(\mathcal{B}\), namely that it must vanish if and only if \(A\) and \(B\) have an eigenstate in common. One way to avoid just mentioned problems is to use another measure of statistical dispersion. The first entropic uncertainty relation was formulated by [PERSON] [11] in 1957 for the position and momentum observables, which was later improved in 1975 by [PERSON] [12] and [PERSON] and [PERSON] [13] resulting in  
\[h(Q)+h(P)\geq\log(e\pi\hbar), \tag{4}\]  
where \(h\) is the differential entropy defined as \(h(Q)=-\int_{\infty}^{\infty}\Gamma(q)\log\Gamma(q)\mathrm{d}q\), with probability density \(\Gamma(q)\). Equation (4) actually implies [PERSON]'s relation Equation (1). The extension to non-degenerate observables on a finite-dimensional Hilbert space was given by [PERSON] in 1983 [14] and later improved by [PERSON] and [PERSON] [15] yielding the well-known entropic uncertainty relation  
\[H(A)+H(B)\geq-\log c,\qquad c:=max_{i,j}|\bra{b_{j}}a_{i}|^{2}, \tag{5}\]  
where \(H\) denotes the Shannon entropy and \(c\) is the maximal overlap between the eigenvectors.