# Double Slit with an Einstein–Podolsky–Rosen Pair

^{1}

^{2}

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^{†}

## Abstract

**:**

## 1. Introduction

## 2. Model for Entangled Double-Slit Experiment Using Gaussian Wave Packets

## 3. A Partially Entangled Double-Slit Experiment

## 4. Model for Asymmetric Entangled Double-Slit Experiment

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Complementary Quantum Measures and Relations

**Definition**

**1.**

**Definition**

**2.**

**Proof.**

**Definition**

**3.**

**Definition**

**4.**

**Proof.**

## Appendix B. Computations and Wigner Functions

#### Appendix B.1. The Wigner Function for a Partially Entangled Double-Slit Experiment

#### Appendix B.2. The Wigner Function for an Asymmetric Double-Slit Experiment

#### Appendix B.3. Purification of Alice’s Subsystem

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**Figure 1.**The proposed “double-double slit” setup: each particle of an entangled two-mode Gaussian state ${\psi}_{\theta}\left(\right)open="("\; close=")">{x}_{1},{x}_{2}$ (5) is sent through a double slit with a screen behind it. The amount of quantum entanglement between the two particles is controlled by a parameter $\theta \in \left(\right)open="["\; close="]">0,\frac{\pi}{4}$. The experiment is repeated many times with the same initial state, and the positions where the particles hit the screen are registered to infer the one-particle and two-particle probability distributions. Generally, the distance between each pair of slits may vary. The plotted probability distributions on the screens are for $\theta =\frac{\pi}{4}$.

**Figure 2.**One-particle visibility ${V}_{\theta}$ (11) and two-particle visibility (predictability) ${P}_{\theta}$ (17) computed from the corresponding quantum probability distributions ${f}_{{p}_{1},\theta}\left({p}_{1}\right)$ (7) and ${\tilde{F}}_{\theta}({p}_{+},{p}_{-}=0)$ (13) for different strengths of entanglement controlled by $\theta =0,\pi /12,\pi /6$ and $\pi /4$. This plot illustrates the smooth transition from zero visibility and maximal predictability, to maximal visibility and zero predictability, as the entanglement diminishes. The bold line depicts a polar curve of ${P}_{\theta}^{2}+{V}_{\theta}^{2}$.

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**MDPI and ACS Style**

Peled, B.Y.; Te’eni, A.; Georgiev, D.; Cohen, E.; Carmi, A.
Double Slit with an Einstein–Podolsky–Rosen Pair. *Appl. Sci.* **2020**, *10*, 792.
https://doi.org/10.3390/app10030792

**AMA Style**

Peled BY, Te’eni A, Georgiev D, Cohen E, Carmi A.
Double Slit with an Einstein–Podolsky–Rosen Pair. *Applied Sciences*. 2020; 10(3):792.
https://doi.org/10.3390/app10030792

**Chicago/Turabian Style**

Peled, Bar Y., Amit Te’eni, Danko Georgiev, Eliahu Cohen, and Avishy Carmi.
2020. "Double Slit with an Einstein–Podolsky–Rosen Pair" *Applied Sciences* 10, no. 3: 792.
https://doi.org/10.3390/app10030792