# Parallel Lives: A Local-Realistic Interpretation of “Nonlocal” Boxes

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## Abstract

**:**

## 1. Introduction

- Principle of realism: There is a real world whose state determines the outcome of all observations.
- Principle of locality: No action taken at some point can have any effect at some remote point at a speed faster than light.

- Any local-realistic world must be described by local hidden variables.
- Quantum theory cannot be described by local hidden variables.
- Ergo, quantum theory cannot be both local and realistic.

## 2. The Imaginary World

#### 2.1. The Nonlocal Box

#### 2.2. Testing the Boxes

- Alice and Bob travel far apart from each other with a large supply of numbered unused boxes, so that Alice’s box number i is the one that is paired with Bob’s box bearing the same number.
- They flip independent unbiased coins labelled 0 and 1 and push the corresponding input buttons on their nonlocal boxes. For each box number, they record the randomly-chosen input and the observed resulting colour. Because they are sufficiently far apart, the experiment can be performed with sufficient simultaneity that Alice’s box cannot know the result of Bob’s coin flip (hence the input to Bob’s box) before it has to flash its own light, and vice versa.
- After many trials, Alice and Bob come back together and verify that the boxes work perfectly: no matter how far they were from each other and how simultaneously the experiment is conducted, the correlations promised in Table 1 are realized for each and every pair of boxes.

#### 2.3. Imperfect Nonlocal Boxes

#### 2.3.1. Quantum Theory and Nonlocal Boxes

## 3. The Many Faces of Locality

#### 3.1. No-Signalling

- Principle of no-signalling: No action taken at some point can have any observable effect at some remote point at a speed faster than light.

#### 3.2. Local Realism Implies No-Signalling

- By the principle of locality, no action taken at point A can have any effect on the state of the world at point B faster than at the speed of light.
- By the principle of realism, anything observable at point B is a function of the state of the world at that point.
- It follows that no action at point A can have an observable effect at point B faster than at the speed of light.

#### 3.3. Local Hidden Variable Theories

#### 3.4. The Einstein–Podolsky–Rosen Argument

- Suppose Alice pushes her input button first. Note that for simplicity, we ignore the fact that there would be no such thing as absolute time if we took account of relativity, so that the notion of who pushes the button first may be ill-defined. This has no impact on the current reasoning because it is well-defined whether the effect of a button push can reach the other side before the other button is pushed.
- When she pushes her button, this cannot have any instantaneous effect on Bob’s box, by the principle of locality.
- After seeing her output, Alice can know with certainty what colour Bob will see as a function of his input (even though she does not know which input he will choose). For example, if Alice had pushed 1 and seen green, she knows that if Bob chooses to push 0 he will also see green, whereas, if he chooses to push 1, he will see red.
- Since it is possible for Alice to know with certainty what colour Bob will see when he pushes either button, and she can obtain this knowledge without influencing his system, it must be that his colour was predetermined as a function of which button he would push. This predetermination can only come from the initial source of shared randomness, and errors could occur if it were influenced by local randomness at Bob’s.

#### 3.5. Local Hidden Variable Theory for Nonlocal Boxes

## 4. Bell’s Theorem

**Theorem**

**1**(Bell’s Theorem)

**.**

**Proof.**

#### Quantum Theory and Bell’s Theorem

## 5. A Local Realistic Solution—Parallel Lives

#### Quantum Theory, Parallel Lives and Many Worlds

## 6. Revisiting Bell’s Theorem and the Einstein–Podolsky–Rosen Argument

#### 6.1. Parallel Lives versus Hidden Variable Theories

#### 6.2. How an Apparent Contradiction Leads to Parallel Lives

- Let us say that Alice pushes button 1 on her box. Without loss of generality, say that her box flashes the green colour.
- Now, we know that Bob will see green if he pushes his button 0, whereas he will see red if he pushes his button 1, according to Table 1. By the principle of locality, this conclusion holds regardless of Alice’s previous action since she was too far for her choice of button to influence Bob’s box.
- What would have happened had Alice pushed her button 0 instead at step 1? She must see the same colour as Bob, regardless of Bob’s choice of button, since her pushing button 0 precludes the possibility that both Alice and Bob will press their button 1, which is the only case yielding different colours, again according to Table 1.
- Statements 2 and 3 imply together that, when Alice pushes her button 0, she must see both red and green!

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

EPR | Einstein–Podolsky–Rosen |

PR | Popescu–Rohrlich |

## Appendix A. Poster on Parallel Lives

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Alice’s Input | Bob’s Input | Output Colours |
---|---|---|

0 | 0 | Identical |

0 | 1 | Identical |

1 | 0 | Identical |

1 | 1 | Different |

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**MDPI and ACS Style**

Brassard, G.; Raymond-Robichaud, P.
Parallel Lives: A Local-Realistic Interpretation of “Nonlocal” Boxes. *Entropy* **2019**, *21*, 87.
https://doi.org/10.3390/e21010087

**AMA Style**

Brassard G, Raymond-Robichaud P.
Parallel Lives: A Local-Realistic Interpretation of “Nonlocal” Boxes. *Entropy*. 2019; 21(1):87.
https://doi.org/10.3390/e21010087

**Chicago/Turabian Style**

Brassard, Gilles, and Paul Raymond-Robichaud.
2019. "Parallel Lives: A Local-Realistic Interpretation of “Nonlocal” Boxes" *Entropy* 21, no. 1: 87.
https://doi.org/10.3390/e21010087