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

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This study provides a rigorous proof for the probability of winning a CHSH game by luck, even with biased or unequal question probabilities. It clarifies the statistical limits for quantum communication security and causal model validity.

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Area of Science:

  • Quantum Information Theory
  • Foundations of Quantum Mechanics
  • Statistical Analysis

Background:

  • Quantum games like the CHSH game demonstrate entanglement's power.
  • Classical strategies limit wins to 75%; higher wins suggest non-classical resources or biases.
  • Finite rounds and unequal probabilities in real games introduce luck and require statistical analysis for applications like eavesdropping detection.

Purpose of the Study:

  • To provide a self-contained proof for the probability of winning a CHSH game by pure luck.
  • To analyze bounds without assuming small biases in random number generators.
  • To investigate unequal probability scenarios in CHSH games.

Main Methods:

  • Developed a fully self-contained proof for CHSH game luck probability bounds.
  • Applied results from McDiarmid and Combes for unequal probability cases.
  • Numerically illustrated exploitable biases in question generation.

Main Results:

  • Established a rigorous bound on the probability of winning a CHSH game by luck.
  • Derived bounds for scenarios with unequal probabilities of question regimes.
  • Demonstrated how biases can be exploited in practical quantum game scenarios.

Conclusions:

  • The study offers a transparent statistical analysis for CHSH game outcomes under various conditions.
  • Findings are crucial for practical applications like quantum communication security and macroscopic Bell tests.
  • Provides a robust framework for analyzing luck-based wins in quantum games with imperfect randomness.