# Measurements of Entropic Uncertainty Relations in Neutron Optics
## 2 Theory
### Entropic Measurement Uncertainty Relation for Qubits  
The principle idea behind the optimization of the uncertainty relation is based on finding the minimum of the sum of entropies \(H(\mathbb{A})+H(\mathbb{B})\). For dichotomic observables there are only two outcomes and the binary Shannon entropy \(h_{b}(x)\) in its symmetric form can be compactly written as  
\[h_{b}(x)=-\tfrac{1+x}{2}\log_{2}\left(\tfrac{1+x}{2}\right)-\tfrac{1-x}{2} \log_{2}\left(\tfrac{1-x}{2}\right)\,\quad x\in[-1,1]. \tag{15}\]  
Let \(A=\vec{a}\cdot\vec{\sigma}\) and \(B=\vec{b}\cdot\vec{\sigma}\) then calculation of the extreme values [36; 37] for qubit systems show that for \(|\vec{a}\cdot\vec{b}|\gtrsim 0.391\) an analytic expression of the uncertainty relation can be provided. According to the details of [32] the noise-noise uncertainty relation is expressed as  
\[g\big{(}N(\mathcal{M},A)\big{)}^{2}+g\big{(}N(\mathcal{M},B)\big{)}^{2}-2|\vec {a}\cdot\vec{b}|\,g\big{(}N(\mathcal{M},A)\big{)}\,g\big{(}N(\mathcal{M},B) \big{)}\leq 1-(\vec{a}\cdot\vec{b})^{2}, \tag{16}\]  
where \(g\) is the inverse function of the binary entropy Equation (15), hence \(g(h_{b}(x))=x,\forall x\in[0,1]\). However for \(|\vec{a}\cdot\vec{b}|\lesssim 0.391\), not all obtainable values by joint measurements are contained in the region demarcated by relation Equation (16). Geometrically, when the modulus of the inner product of the two Bloch vectors \(|\vec{a}\cdot\vec{b}|\) is larger than the critical angle \(\approx 67^{\circ}\) the shape of the area becomes non-convex. According to the proposition [32] the entire region of obtainable values for qubits is given by the convex hull of the subset constituted by Equation (16), i.e.,  
\[R_{NN}(A,B)=\text{conv}\big{\{}(s,t)\ |\ g(s)^{2}+g(t)^{2}-2|\vec{a}\cdot \vec{b}|\,g(s)\,g(t)\leq 1-(\vec{a}\cdot\vec{b})^{2}\big{\}}. \tag{17}\]  
Let \(P_{\pm}(\vec{r}_{i})=\tfrac{1}{2}(1\pm\vec{r}_{i}\cdot\vec{\sigma})\) be two-dimensional, orthogonal projectors, where the Bloch sphere coordinates are used to parametrize \(\vec{r}_{i}=\vec{r}_{i}(\theta_{i},\phi_{i})\) by the polar \(\theta_{i}\) and azimuthal angle \(\phi_{i}\). The entire region including the lower boundary of \(R_{NN}(A,B)\) can be obtained by measuring four-outcome POVMs corresponding to a statistical mixture of two projectors weighted by \(q\) and \(1-q\),  
\[M_{m}=\{qP_{+}(\vec{r}_{1}),qP_{-}(\vec{r}_{1}),(1-q)P_{+}(\vec{r}_{2}),(1-q)P_ {-}(\vec{r}_{2})\}. \tag{18}\]  
A generally valid and optimal uncertainty relation between the entropic noise \(N(\mathcal{M},A)\) and the disturbance \(D(\mathcal{M},B)\) has not been found so far, not even for two-dimensional quantum systems.  
At least for two Pauli observables \(\sigma_{i},\sigma_{j}\) which are orthogonal \(\text{Tr}(\sigma_{i}\sigma_{j})=0\) and a measurement apparatus that performs projective measurements \(\mathcal{M}=\{P_{+}(\vec{r}),P_{-}(\vec{r})\}\) the following inequality does hold  
\[g(N(\mathcal{M},\sigma_{i}))^{2}+g(D(\mathcal{M},\sigma_{j}))^{2}\leq 1. \tag{19}\]  
Apart from this special case, an entropic uncertainty relation for \(N(\mathcal{M},A)\) - \(D(\mathcal{M},B)\) has yet to be found. For a detailed survey and conjectures see [32].