# Uncertainty Relation between Detection Probability and Energy Fluctuations

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model and Notation

## 3. Lower Bound Using the Propagator

**Remark**

**1.**

**Remark**

**2.**

## 4. Uncertainty Relation

**Remark**

**3.**

**Remark**

**4.**

**Remark**

**5.**

## 5. The Reverse Dark Approach

## 6. Further Improvement of the Uncertainty Relation

## 7. Quantum Walks on Graphs

#### 7.1. Finite Line

**Remark**

**6.**

**Remark**

**7.**

#### 7.2. Enumeration of Paths Approach

#### 7.3. The Benzene-Like Ring

#### 7.4. Optimisations of the Lower Bound

#### 7.5. Other Examples

## 8. Discussion

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**A quantum walk on a line with five sites. The initial condition is any localised state denoted with circles with an interior coloured green. The detected state $|\mathrm{d}\rangle $ is the circle with a white interior. Here we present the notation used in the text, the exact results of ${P}_{\mathrm{det}}$ and the lower bound found using the uncertainty relation. For example, starting on $|1\rangle $ and measuring on $|2\rangle $ we have ${P}_{\mathrm{det}}=2/3$ while the uncertainty relation gives ${P}_{\mathrm{det}}\ge 1/2$.

**Figure 2.**For a quantum walk starting on a node of the graph and measured elsewhere, we present the exact result for the detection probability, which depend’s on the location of the starting point. Shown also is the lower bound, obtained with the uncertainty principle. We choose s as the smallest integer, for which the uncertainty principle is non-trivial, namely the case where $\langle {\psi}_{\mathrm{in}}|{H}^{s}|\mathrm{d}\rangle \ne 0$, this is the distance between the measured site and the initial condition.

**Figure 3.**A system with a dangling bond, where the edge state $|0\rangle $ is repeatedly measured. Note that in the absence of the bond, i.e., when node $|6\rangle $ is removed, the detection probability is unity, no matter what is the starting point. To obtain the lower bound, using uncertainty, we choose s to be the shortest distance between the initial condition and the detected state. In Figure 4 we improve the bound for the transition $|5\rangle \to |0\rangle $, which here gives ${P}_{\mathrm{det}}\ge 1/78$.

**Figure 4.**We demonstrate the optimisation of the uncertainty principle, for a system with a dangling bond. The Figure shows the lower bound for ${P}_{\mathrm{det}}$ versus s. The measurement is on node $|0\rangle $ and we consider two initial conditions: one localised $|{\psi}_{\mathrm{in}}\rangle =|5\rangle $ and the other uniform. See upper part of Figure 3 for schematics and notation. Starting on node $|5\rangle $ and when $s\le 4$ the bound is equal to zero since if the initial and detected states are localised, s must be larger than the distance between these states to make the approach useful. Increasing s clearly leads to improvements of the bound. The exact results are ${P}_{\mathrm{det}}=20/21$ for the uniform initial condition, and ${P}_{\mathrm{det}}=2/3$ for the initially localised state.

**Table 1.**For the system with a dangling bond, the detection being on $|0\rangle $, the Table gives the lower bound for the uniform initial condition versus s. Increasing s in this range is improving the bound.

s | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

${P}_{\mathrm{det}}\ge $ | $\frac{2}{7}$ | $\frac{2}{7}$ | $\frac{11}{21}$ | $\frac{42}{119}$ | $\frac{167}{253}$ |

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**MDPI and ACS Style**

Thiel, F.; Mualem, I.; Kessler, D.; Barkai, E. Uncertainty Relation between Detection Probability and Energy Fluctuations. *Entropy* **2021**, *23*, 595.
https://doi.org/10.3390/e23050595

**AMA Style**

Thiel F, Mualem I, Kessler D, Barkai E. Uncertainty Relation between Detection Probability and Energy Fluctuations. *Entropy*. 2021; 23(5):595.
https://doi.org/10.3390/e23050595

**Chicago/Turabian Style**

Thiel, Felix, Itay Mualem, David Kessler, and Eli Barkai. 2021. "Uncertainty Relation between Detection Probability and Energy Fluctuations" *Entropy* 23, no. 5: 595.
https://doi.org/10.3390/e23050595