# Measurements of Entropic Uncertainty Relations in Neutron Optics
## 1 Introduction  
According to quantum mechanics, any single observable or a set of compatible observables can be measured with arbitrary accuracy. However, the physical scenario changes when jointly (i.e., simultaneously or successively) measuring non-commuting observables. [PERSON]'s seminal paper from 1927 [1] predicts a lower bound on the uncertainty of a joint measurement of incompatible observables, more precisely of the canonically conjugate variables position and momentum. On the other hand, it also sets an upper bound on the accuracy with which the values of non-commuting observables can be simultaneously prepared. While in the past these two statements have often been mixed, they are now clearly distinguished as measurement uncertainty and preparation uncertainty relations. When we consider the quantities position and momentum, as in [PERSON]'s original argument, the well-known inequality states that it is impossible to prepare a quantum system for which both, position and momentum, are arbitrarily sharply defined. Operationally this is achieved by an apparatus that prepares many copies of a quantum particle in the same way, which is schematically illustrated in Figure 1. On one part of the ensemble, the position is measured and on the other part the momentum. From the statistical distribution of these two measurements, the preparation uncertainty can be determined. What is the correct way to interpret these distributions? While [PERSON]'s paper only presented his ideaheuristically, the first rigorously-proven uncertainty relation for position \(Q\) and momentum \(P\) was provided by [PERSON] [2] in terms of standard deviations expressed as  
\[\Delta(Q)\Delta(P)\geq\frac{\hbar}{2}, \tag{1}\]  
where the standard deviation is defined as \(\Delta(A)^{2}=\langle\psi|A^{2}|\psi\rangle-\langle\psi|A|\psi\rangle^{2}\). This relation can be generalized to other pairs of incompatible observables, including the phase and excitation number of a harmonic oscillator, the angle and the orbital angular momentum of a particle, and orthogonal components of spin angular momentum [3; 4; 5]. In 1929, [PERSON] [6] extended [PERSON]'s relation Equation (1) to an arbitrary pair of observables \(A\) and \(B\) as  
\[\Delta(A)\Delta(B)\geq|\frac{1}{2i}\langle\psi|[A,B]|\psi\rangle|, \tag{2}\]  
with the commutator \([A,B]=AB-BA\). [PERSON]'s uncertainty relation in turn follows from a slightly stronger inequality namely the Schrodinger uncertainty relation [7] which is given by  
\[\Delta(A)\Delta(B)\geq|\langle\psi|\{A,B\}|\psi\rangle-\langle\psi|A|\psi \rangle\langle\psi|B|\psi\rangle|^{2}+|\frac{1}{2i}\langle\psi|[A,B]|\psi \rangle|^{2}, \tag{3}\]  
introducing the anti-commutator \(\{A,B\}=AB+BA\). In [PERSON]'s and [PERSON]'s uncertainty relations Equations (1) and (2) the uncertainty is expressed in terms of standard deviations. This is, however, not the only way to express uncertainties and sometimes not a well-suited measure [8]. For illustration consider the well-known single-slit experiment, which is schematically illustrated in Figure 2a. An incoming monochromatic plane wave represents an incoming beam of particles. The particles with momentum \(p_{0}\) pass a slit of width \(2a\) and are detected on a screen (or photographic plate) placed at a distance \(l\). The diffraction of particles by such a single slit is frequently used as an illustration of the uncertainty principle [9; 10]. The wave function at the screen (\(z=0\)) may be represented by the following Fourier transform pair illustrated in Figure 2b (for \(l\gg a\gg l/p_{o}\) and \(l\gg|q|\) this is a good approximation of a Fourier transform). The uncertainty in momentum \(\Delta p\) corresponds to the width of the central peak, which contains about 95 of the total probability. Therefore \(\Delta p\) is a very reasonable measure of uncertainty.