# Measurements of Entropic Uncertainty Relations in Neutron Optics

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## Abstract

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## 1. Introduction

## 2. Theory

- Noise–noise uncertainty relation$$N(\mathcal{M},A)+N(\mathcal{M},B)\ge -log(\underset{i,j}{max}|\langle {a}_{i}|{b}_{j}\rangle {|}^{2}).$$This tradeoff expresses the inability for the device $\mathcal{M}$ to jointly discriminate the eigenstates $|{a}_{i}\rangle $ and $|{b}_{i}\rangle $ to arbitrary precision.
- Noise–disturbance uncertainty relation$$N(\mathcal{M},A)+{D}_{\mathcal{E}}(\mathcal{M},B)\ge -log(\underset{i,j}{max}{\left|\langle {a}_{i}|{b}_{j}\rangle \right|}^{2}).$$This inequality implies that the measurement apparatus cannot be noise and disturbance-free for both eigenstates of the non-commuting operators A and B.

#### Entropic Measurement Uncertainty Relation for Qubits

## 3. Measurement of Entropic Noise–Noise Uncertainty Relation

#### 3.1. Set-Up

#### 3.2. Measurement Results

## 4. Measurement of Entropic Noise–Disturbance Uncertainty Relation

#### 4.1. Set-Up

#### 4.2. Measurement Results

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**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.

**Figure 2.**Experimental procedure to measure preparation uncertainty relations, where each particle undergoes a measurement of observable A or B. (

**a**) schematic illustration of the single-slit configuration. (

**b**) Distributions of the canonical conjugated variables position q and momentum p.

**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.

**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 ($|\pm a\rangle ,|\pm b\rangle $) uniformly at random and DC-Coil 3 adjusted to make a selection of the state with orientation ${\overrightarrow{r}}_{1}({\theta}_{1})$. 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 $q={\left|\langle +z|+\psi \rangle \right|}^{2}={\left|\langle +z|{U}_{DC1}|+z\rangle \right|}^{2}$. Whenever the experiment is set to q, Analyzer 1 makes a projection onto $P({\overrightarrow{r}}_{1}({\theta}_{1}))$, but when the probability of transmission is inverted to $1-q={\left|\langle +z|\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\psi \rangle \right|}^{2}$, the second stage is conditioned to analyze $P({\overrightarrow{r}}_{2}({\theta}_{2}))$.

**Figure 5.**The neutron count rate for (

**a**) ${I}_{+a,m}$ and (

**b**) ${I}_{+b,m}$ of the four-outcome POVM at a probability weighting of $q=1$ in the case $A={\sigma}_{z}$ and $B={\sigma}_{y}$. Each histogram illustrates the change of counted neutrons depending on the angle $\theta $ of the measurement direction ${\overrightarrow{r}}_{1}={(0,sin{\theta}_{1},cos{\theta}_{1})}^{T}$ with an increment of $\Delta {\theta}_{1}={20}^{\circ}$. The maximal height in the 60s measurement are approximately 2500. At the initial angle ${\theta}_{1}=0$, outcome $m=1$ corresponds to the projector ${P}_{+}({\overrightarrow{r}}_{1})={P}_{+}(\overrightarrow{a})$ and therefore a maximal count rate occurs in (

**a**), while $m=2$ is associated with ${P}_{-}({\overrightarrow{r}}_{1})={P}_{-}(\overrightarrow{a})$ and no neutrons are detected. A rotation of the projectors by angle $\theta $ leads to a decrease of counts in $m=1$ and an increase of measured events in $m=2$ for the input state $|+a\rangle $. Likewise, (

**b**) shows the histogram for an input state $|+b\rangle $, which is a distribution phase-shifted to ${I}_{+a,m}$ by $\theta ={90}^{\circ}$. Outcomes $m=3$ and $m=4$ from Equation (18) do not occur in the projective limit $q=1$.

**Figure 6.**The neutron count rate for (

**a**) ${I}_{+a,m}$ and (

**b**) ${I}_{+b,m}$ of the four-outcome POVM at polar angles ${\theta}_{1}={0}^{\circ}$ and ${\theta}_{2}={90}^{\circ}$ in the case $A={\sigma}_{z}$ and $B={\sigma}_{y}$. Both histograms display a variation in the count rate depending on the probability weighting q with an approximate change rate of $\Delta q=0.1$. In (

**a**) the measurement start with the projective case $q=1$ and therefore $|+a\rangle $ is measured perfectly by ${P}_{+}({\overrightarrow{r}}_{1})={P}_{+}(\overrightarrow{a})$. Alternation of the statistical mixture means that outcomes $m=3$ and $m=4$ measuring in direction ${\overrightarrow{r}}_{2}=\overrightarrow{b}$ contribute more and more to the statistics the more q decreases. As for (

**b**), the measurement of A with the eigenstate of $|+b\rangle $ starts with half the count rate and then decreases equiprobably the more q is decreased. Meanwhile $m=3$ corresponing to $(1-q){P}_{+}({\overrightarrow{r}}_{2})=(1-q){P}_{+}(\overrightarrow{b})$ registers more neutrons the closer q gets to 0, while $m=4$ associated with $(1-q){P}_{-}({\overrightarrow{r}}_{2})=(1-q){P}_{-}(\overrightarrow{b})$ remains practically zero for the input state $|+b\rangle $.

**Figure 7.**The measurement results of noises $N(\mathcal{M},A)$ in blue and $N(\mathcal{M},B)$ in red for $A={\sigma}_{z}$ and $B\simeq sin({79}^{\circ}){\sigma}_{y}+cos({79}^{\circ}){\sigma}_{z}$ together with the theoretical predictions given by the solid curves in the projective case. In addition, the purple data points show how variation of the azimuthal angle ${\varphi}_{1}$, which moves the measurement operator ${P}_{\pm}({\overrightarrow{r}}_{1})$ out of the plane spanned by $\overrightarrow{a}-\overrightarrow{b}$, increases the total noise and makes the noise-noise trade-off worse. The black curve is equal to the sum $N(\mathcal{M},A)+N(\mathcal{M},B)$ and has two notable minima in the interval $0\le {\theta}_{1}<\pi $, which are necessary to identify the optimal values for the POVM.

**Figure 8.**Theoretical lines plus measurement points of $N(\mathcal{M},A)$ in blue and $N(\mathcal{M},B)$ in red for $A={\sigma}_{z}$ and $B\simeq sin({79}^{\circ}){\sigma}_{y}+cos({79}^{\circ}){\sigma}_{z}$ for varying statistical weighting q. To obtain this plot the operators ${P}_{\pm}({\overrightarrow{r}}_{1})$ and ${P}_{\pm}({\overrightarrow{r}}_{2})$ are fixed to the extreme values ${\theta}_{1}={5}^{\circ}$ and ${\theta}_{2}={74}^{\circ}$ corresponding to the minima in Figure 7. The black line shows that the sum of the entropy remains constant at the lowest value of Figure 7. As a result, realizing the POVM Equation (18) reduces the overall noise.

**Figure 9.**Parametric plots of noises in observable $A=\overrightarrow{a}\xb7\overrightarrow{\sigma}$ and $B=\overrightarrow{b}\xb7\overrightarrow{\sigma}$ which indicate the accessible joint measurement uncertainty region ${R}_{NN}(A,B)$, with the blue curve being the result of projective measurements in the plane spanned by $\overrightarrow{a}$-$\overrightarrow{b}$, the orange colored region obtainable by arbitrary projectors and finally the green area demarcating the lowest pairs of noise values only reached by POVMs. The upper panel shows the improvement on the uncertainty relation by choosing the convex combination of the operators ${P}_{\pm}({\overrightarrow{r}}_{1})$ and ${P}_{\pm}({\overrightarrow{r}}_{2})$ controlled by q, whose color coding is listed for (

**a**) $\overrightarrow{a}\sphericalangle \overrightarrow{b}\simeq {90}^{\circ}$, (

**b**) $\overrightarrow{a}\sphericalangle \overrightarrow{b}\simeq {85}^{\circ}$ and (

**c**) $\overrightarrow{a}\sphericalangle \overrightarrow{b}\simeq {79}^{\circ}$. In the lower panel, the included angle is very close or below the critical value of $\approx {67}^{\circ}$. Apparently the orange region is already becoming convex and further improvements by POVMs are not feasible for (

**d**) $\overrightarrow{a}\sphericalangle \overrightarrow{b}\simeq {69.5}^{\circ}$ or possible for (

**e**) $\overrightarrow{a}\sphericalangle \overrightarrow{b}\simeq {60}^{\circ}$.

**Figure 10.**Experimental realization for the measurement of noise $N(\mathcal{M},A)$ and disturbance ${D}_{\mathcal{E}}(\mathcal{M},B)$. In the preparation phase, the monochromatic neutrons are transformed to the observables’ eigenstates, indicated by the left Bloch sphere. DC-2 controls the incident angle $\theta $ of the spin. As noted in the middle Bloch sphere, the analyzer transmits best in the z-direction and gets worse as the operator $M=sin\theta \phantom{\rule{0.166667em}{0ex}}{\sigma}_{y}+cos\theta \phantom{\rule{0.166667em}{0ex}}{\sigma}_{z}$ turns away from the poles. DC-3 can either be used to produce the uncorrected eigenstates $|\pm m\rangle $ or perform the optimal correction and generate the spins $|\pm b\rangle =|\pm y\rangle $. This different mode of operation is characterized by the light green color mark. In the dark green area, the second, subsequent projection takes place, in which DC-4 transforms the incoming vectors in such a way that B is measured optimally.

**Figure 11.**Measurement results of the conditional probabilities for (a) $p(m|a)$, (b) $p({b}^{\prime}|b)$ and (c) ${p}_{corr}({b}^{\prime}|b)$. From $\theta =0$ to $\theta =\pi /2$ the increment is $\pi /18$ and changes to $\pi /9$ from $5\pi /9$ on. At the angle $\theta =0$ the measurement operator is ${\sigma}_{z}$ and the probability to receive $m=\pm 1$ given the value of the initial eigenvector is $a=\pm 1$ is maximal in (

**a**), while the results with opposite signs $a=\mp 1$ are impossible to occur. Rotating the measurement direction leads to a sinusoidal change of the probabilities. The center (

**b**) and right graph (

**c**) show the probabilities for receiving ${b}^{\prime}$ given state b. The maxima and minima are shifted in phase to the first panel by $\pi /2$. The uncorrected curve in the middle has not the same wide arc form compared to the curve where the optimal correction procedure is conducted.

**Figure 12.**Entropic noise $N(\mathcal{M},A)$ in blue and uncorrected/optimally corrected disturbance ${D}_{\mathcal{E}}(\mathcal{M},B)$ in green/red as a function of the angle $\theta $ of the measurement operator. The noise for observable $A={\sigma}_{z}$ at $\theta =0$ is minimal, while the eigenstates of $|\pm b\rangle $ are maximally disturbed at this angle. Rotating the polar angle closer to the observable $B={\sigma}_{y}$ reduces the disturbance at the expanse of increasing the noise. After the first apparatus an additional rotation of the output state $|m\rangle $ onto the closest eigenvector of B reduces the loss of correlation in parts and reduces the disturbance furthermore. At $\theta =\pi /2$ the measurement apparatus makes a perfect measurement of B inducing no disturbance, being however maximally noisy with respect to a measurement of $|\pm a\rangle $.

**Figure 13.**Noise disturbance uncertainty relation for projective measurements for the uncorrected case in red and the corrected set-up in green. The data points are either on the green curve Equation (19) or above. It is apparent that the generally predicted Equation (12), given by the black dashed line, is not saturated for qubits except for the case where either noise or disturbance is maximal.

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**MDPI and ACS Style**

Demirel, B.; Sponar, S.; Hasegawa, Y.
Measurements of Entropic Uncertainty Relations in Neutron Optics. *Appl. Sci.* **2020**, *10*, 1087.
https://doi.org/10.3390/app10031087

**AMA Style**

Demirel B, Sponar S, Hasegawa Y.
Measurements of Entropic Uncertainty Relations in Neutron Optics. *Applied Sciences*. 2020; 10(3):1087.
https://doi.org/10.3390/app10031087

**Chicago/Turabian Style**

Demirel, Bülent, Stephan Sponar, and Yuji Hasegawa.
2020. "Measurements of Entropic Uncertainty Relations in Neutron Optics" *Applied Sciences* 10, no. 3: 1087.
https://doi.org/10.3390/app10031087