# From Quantum Probabilities to Quantum Amplitudes

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## Abstract

**:**

## 1. Introduction

## 2. From Amplitudes to Probabilities

## 3. Von Neumann Measurements and the Two-Step Histories

## 4. From Probabilities to Amplitudes. Three-Step Histories

## 5. An Inverse Measurement Problem

## 6. Double-Slit Interference

## 7. A Simple Example

## 8. Accurate (Strong) and Inaccurate (Weak) Limits

## 9. Averages and the “Weak Measurements”

## 10. Prediction and Retrodiction

## 11. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Derivation of Equation (23)

## Appendix B. The Factors $\mathcal{F}$_{0}, $\mathcal{F}$_{1}, and $\mathcal{L}$ in Equations (50) and (51)

## References

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**Figure 1.**Four virtual paths for a two-level system. The times $t=0$ and $t={t}^{\prime}$ belong to the “past”, while $t={t}^{\u2033}$ refers to the “present”, and must be treated differently.

**Figure 2.**With only two amplitudes, ${A}^{s}({d}_{k}\leftarrow {c}_{1}\leftarrow {b}_{i})$ and ${A}^{s}({d}_{k}\leftarrow {c}_{2}\leftarrow {b}_{i})$ to be evaluated, the “spin”’s state $|Z\rangle $ in Equation (31) lies on a Bloch sphere. If the pointer has no own dynamics, e.g., $M\to \infty $, evaluating only its position (momentum) distribution leaves the azimuthal angle $\varphi $ (polar angle $\theta $) indeterminate. The problem is remedied if the pointer’s state is allowed to spread, as discussed in Section 7.

**Figure 3.**Normalised probability density, ${\rho}^{\prime}\left(f\right)=\rho \left(f\right)/\int \rho \left({f}^{\prime}\right)d{f}^{\prime}$, of the pointer’s readings for a two-level system making a transition between the initial and final states (41). Dividing the range of f into four equally probable regions ensures the same accuracy in approximating the probabilities by the relative frequencies in the r.h.s. of Equation (39).

**Figure 4.**The moduli and relative phase of the amplitudes ${A}^{s}({d}_{k}\leftarrow {c}_{j}\leftarrow {b}_{i})\equiv \left|{A}_{j}\right|exp\left(i{\varphi}_{j}\right)$, $j=1,2$, as function of the number of the successful post-selection, K, evaluated as K increases by 100. The dashed lines indicate the exact values.

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Martínez-Garaot, S.; Pons, M.; Sokolovski, D.
From Quantum Probabilities to Quantum Amplitudes. *Entropy* **2020**, *22*, 1389.
https://doi.org/10.3390/e22121389

**AMA Style**

Martínez-Garaot S, Pons M, Sokolovski D.
From Quantum Probabilities to Quantum Amplitudes. *Entropy*. 2020; 22(12):1389.
https://doi.org/10.3390/e22121389

**Chicago/Turabian Style**

Martínez-Garaot, Sofia, Marisa Pons, and Dmitri Sokolovski.
2020. "From Quantum Probabilities to Quantum Amplitudes" *Entropy* 22, no. 12: 1389.
https://doi.org/10.3390/e22121389