# Winning a CHSH Game without Entangled Particles in a Finite Number of Biased Rounds: How Much Luck Is Needed?

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## Abstract

**:**

## 1. Introduction

## 2. The CHSH Game

#### 2.1. The Game with the Goal of Achieving a High Percentage of Winning Rounds

#### 2.2. The Game with the Goal of Achieving a High S-Value

#### 2.3. On the Equivalence of the Two Game Descriptions

## 3. Bounds for the Likelihood of Being Lucky

- (i)
- Draw up a list, which specifies in advance which elementary strategy they will use in a game round only based on the question posed in this round;
- (ii)
- Create a deterministic algorithm that determines how they will answer in a given round based on the information about all questions they have received so far locally;
- (iii)
- Use the information about all answers they have given so far locally and all questions they have received so far locally as an input to an algorithm, which may use classical local independent randomization procedures while the game is ongoing, to generate their answers;
- (iv)
- Determine how they will answer in a given round using a momentary hunch (i.e., gut feeling) while the game is ongoing.

#### 3.1. Bounds Based on Hoeffding Inequalities

**Proposition**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

#### 3.2. Bounds Based on an Extended McDiarmid Inequality

**Proposition**

**2.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 4. Monte Carlo Simulations and Exploitable Biases

## 5. Discussion and Summary

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Two probability distributions for ${S}_{1}^{\mathrm{obs}}$ generated by a Monte Carlo simulation of $10,000$ CHSH games of $N=500$ rounds each. The threshold of $2+\eta =2.25$ is shown in red. (

**left**) The graph on the left-hand side was generated with Alice and Bob randomly picking elementary strategies with $L=1$, while the regimes $xy$ were generated by independent and unbiased coin tosses. The simulated probability is $Pr\{{S}_{1}^{\mathrm{obs}}\u2a7e2.25\}=4.5\%$ with a maximum value of ${S}_{1,max}^{\mathrm{obs}}=2.52$ observed in a single CHSH game. (

**right**) The graph on the right-hand side has been generated with Alice and Bob randomly picking elementary strategies that win in the regime $xy=00$ as well as satisfy $L=1$. Here, all regimes $xy$ were generated by independent, but biased coin tosses with $P(xy=00)=0.7$ and $P(xy=01)=P(xy=10)=P(xy=11)=0.1$. The simulated probability is $Pr\{{S}_{1}^{\mathrm{obs}}\u2a7e2.25\}=14\%$ with a maximum value of ${S}_{1,max}^{\mathrm{obs}}=2.86$ observed in a single CHSH game.

**Table 1.**The following table illustrates the goal implied by Equation (1) for the four possible question regimes and it lists the elementary strategies that result in a loss under each question regime. Instead of showing all four bits of a losing elementary strategy $({A}_{0},{A}_{1},{B}_{0},{B}_{1})$, only the bits corresponding to the relevant question regime are shown, because the value of the other bits is irrelevant. Note that, in each round, Alice only knows x and has to answer by giving the value of a, while Bob only knows y and has to give the answer b.

x | y | Regime | Needed to Win | Losing Strategies |
---|---|---|---|---|

0 | 0 | $xy=00$ | $0=a\oplus b$ | $({A}_{0},{B}_{0})=(0,1)$ or $(1,0)$ |

0 | 1 | $xy=01$ | $0=a\oplus b$ | $({A}_{0},{B}_{1})=(0,1)$ or $(1,0)$ |

1 | 0 | $xy=10$ | $0=a\oplus b$ | $({A}_{1},{B}_{0})=(0,1)$ or $(1,0)$ |

1 | 1 | $xy=11$ | $1=a\oplus b$ | $({A}_{1},{B}_{1})=(0,0)$ or $(1,1)$ |

**Table 2.**The following table illustrates the bounds for the certainty that the CHSH game is not won by chance, assuming four equally probable question regimes. Column (a) illustrates the bound obtained with Equation (12), Column (b) illustrates the best available bounds from Theorem 1 and Column (c) illustrates the best available bounds from Theorem 2.

N | $\mathit{\eta}$ | (a) | (b) | (c) |
---|---|---|---|---|

500 | $0.5$ | negative | negative | $37.73\%$ |

1000 | $0.5$ | negative | $13.51\%$ | $69.66\%$ |

2500 | $0.5$ | $30.37\%$ | $93.33\%$ | $97.23\%$ |

5000 | $0.5$ | $93.94\%$ | $99.90\%$ | $99.95\%$ |

**Table 3.**The following table illustrates the bounds for the certainty that the CHSH game is not won by chance over N rounds with threshold $\eta $ assuming unequal probabilities for the four question regimes. Columns (a) and (c) illustrate the bounds obtained from Theorem 1 and Columns (b) and (d) the bounds from Theorem 2. The values in Columns (a) and (b) were computed for a weakly asymmetric situation with ${p}_{00}=22\%$ and ${p}_{01}={p}_{10}={p}_{11}=26\%$. The values in Columns (c) and (d) were computed for a strongly asymmetric situation with ${p}_{00}=10\%$ and ${p}_{01}={p}_{10}={p}_{11}=30\%$.

N | $\mathit{\eta}$ | (a) | (b) | (c) | (d) |
---|---|---|---|---|---|

500 | $0.5$ | negative | $30.91\%$ | negative | $4.04\%$ |

1000 | $0.5$ | $12.71\%$ | $60.45\%$ | negative | $12.35\%$ |

2500 | $0.5$ | $92.91\%$ | $93.82\%$ | $69.61\%$ | $37.20\%$ |

5000 | $0.5$ | $99.87\%$ | $99.75\%$ | $93.97\%$ | $66.35\%$ |

500 | $0.75$ | $14.51\%$ | $57.42\%$ | negative | $9.36\%$ |

1000 | $0.75$ | $85.39\%$ | $87.98\%$ | $52.34\%$ | $26.32\%$ |

2500 | $0.75$ | $99.92\%$ | $99.79\%$ | $93.37\%$ | $65.46\%$ |

5000 | $0.75$ | $99.99\%$ | $99.99\%$ | $99.73\%$ | $91.58\%$ |

**Table 4.**An example for a systematic cyclical bias in the random question regimes that repeats over the round numbers n. The number k is understood to run through the natural numbers $k=1,2,3,\dots $.

Round n | $\mathit{P}(\mathit{xy}=00)$ | $\mathit{P}(\mathit{xy}=01)$ | $\mathit{P}(\mathit{xy}=10)$ | $\mathit{P}(\mathit{xy}=11)$ |
---|---|---|---|---|

⋯ | ||||

$4k$ | c | $(1-c)/3$ | $(1-c)/3$ | $(1-c)/3$ |

$4k+1$ | $(1-c)/3$ | c | $(1-c)/3$ | $(1-c)/3$ |

$4k+2$ | $(1-c)/3$ | $(1-c)/3$ | c | $(1-c)/3$ |

$4k+3$ | $(1-c)/3$ | $(1-c)/3$ | $(1-c)/3$ | c |

⋯ |

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**MDPI and ACS Style**

Gallus, C.; Blasiak, P.; Pothos, E.M.
Winning a CHSH Game without Entangled Particles in a Finite Number of Biased Rounds: How Much Luck Is Needed? *Entropy* **2023**, *25*, 824.
https://doi.org/10.3390/e25050824

**AMA Style**

Gallus C, Blasiak P, Pothos EM.
Winning a CHSH Game without Entangled Particles in a Finite Number of Biased Rounds: How Much Luck Is Needed? *Entropy*. 2023; 25(5):824.
https://doi.org/10.3390/e25050824

**Chicago/Turabian Style**

Gallus, Christoph, Pawel Blasiak, and Emmanuel M. Pothos.
2023. "Winning a CHSH Game without Entangled Particles in a Finite Number of Biased Rounds: How Much Luck Is Needed?" *Entropy* 25, no. 5: 824.
https://doi.org/10.3390/e25050824