# Measurements of Entropic Uncertainty Relations in Neutron Optics
## 3 Measurement of Entropic Noise-Noise Uncertainty Relation  
### Set-Up  
The experiments presented here use neutrons generated by a nuclear reaction at the TRIGA Mark II reactor in Vienna. The neutrons, after being monochromatized, arrive at the experimental site through an outlet of the shielding in the reactor wall. A polarizer at the beginning of the beamline eliminates one spin component, leaving only monoenergetic neutrons with up spin. A schematic view of the experimental setup is depicted in Figure 4.  
Neutron spins can be manipulated precisely with static magnetic field generated by solenoid coils. When a neutron propagates in the \(y\)-direction and has a spin-up \(\left|+z\right\rangle\), then a magnet field in the perpendicular \(x\)-direction creates a unitary Larmor precession \(U_{DC}\) by an angle \(\alpha\)  
\[U_{DC}\left|+z\right\rangle=e^{i\alpha\sigma_{x}}\left|+z\right\rangle=\cos \frac{\alpha}{2}\left|+z\right\rangle+i\sin\frac{\alpha}{2}\left|-z\right\rangle. \tag{20}\]  
The angle of rotation is proportional to the magnetic field strength and the time of passage of the neutron through the coil \(\alpha=\gamma B_{x}t\), \(\gamma\) being the gyromagnetic ratio. Since the transition time \(t\) is  
Figure 4: Experimental arrangement for determining the noises and thus the uncertainty relation. In (**a**) the full set-up is shown. In the projective case \(q=1\) in (**b**), the configuration consists of 2 coils, DC-Coil 2 generating one of the possible eigenstates (\(\left|\pm a\right\rangle,\left|\pm b\right\rangle\)) uniformly at random and DC-Coil 3 adjusted to make a selection of the state with orientation \(\vec{r}_{1}(\theta_{1})\). For the positive operator-valued measure (POVM), the four-outcomes are implemented by ‘splitting’ the total neutron counts to two partitions in the rose colored area. For that purpose, the DC-Coil 1 changes the angle incident to Analyzer 1 and reduces the probability to be transmitted to the experimental backstage to \(q=\left|\left\langle+z\right|+\psi\right\rangle\left|^{2}=\left|\left\langle+ z\right|U_{DC1}+z\right\rangle\left|^{2}\right\rangle\). Whenever the experiment is set to \(q\), Analyzer 1 makes a projection onto \(P(\vec{r}_{1}(\theta_{1}))\), but when the probability of transmission is inverted to \(1-q=\left|\left\langle+z\right|-\psi\right\rangle\left|^{2}\right\rangle\), the second stage is conditioned to analyze \(P(\vec{r}_{2}(\theta_{2}))\). constant, the spin state is entirely controlled by the electric current that generates the magnetic field \(B_{x}\). Because the neutron is sensitive to magnetic fields, there are also unwanted disturbances from the earth's magnetic field or magnetic materials in the vicinity of the setup. To avoid irritations of the neutron spin, the entire beamline is surrounded by a 13G vertical guide field. In order to leave the field's axis in the coils unchanged in \(x\)-direction, the external field is compensated by a local field in the opposite direction. A supermirror analyzer makes a spin-dependent projection by taking in the input quantum system \(|\psi\rangle\) and producing the outcome \(|+z\rangle\). The use of a coil placed in front of an analyzer, working according to Equation (20), allows any state to be generated before entry. A coil after the analyzer, in turn, allows an arbitrary initial state to be re-established. A measurement in the analyzer, therefore, consists of a combination of coil and supermirror, the usual sequence is the following: the state \(|\psi\rangle=U_{\text{DC1}}\ |+z\rangle\) is generated by DC-Coil 1 and fed to the analyzer 1, the outcome \(|+z\rangle\) appears with probability \(|\langle+z|\psi\rangle|^{2}\), which in turn is transformed by DC-Coil 2 after the analyzer to \(U_{\text{DC2}}\ |+z\rangle\). The idea for the realization of the POVM Equation (18) is as follows.