Invertible Map between Bell Nonlocal and Contextuality Scenarios.

Area of Science:

• Quantum information theory
• Foundations of quantum mechanics
• Computational complexity theory

Background:

• Bell nonlocality demonstrates quantum correlations violate local realism.
• Contextuality describes how quantum systems depend on measurement context.
• Understanding the relationship between these phenomena is crucial for quantum foundations.

Purpose of the Study:

• To establish a formal connection between Bell correlations and contextuality behaviors.
• To investigate the properties of the set of quantum contextual behaviors.
• To explore the computational complexity of identifying quantum contextual behaviors.

Main Methods:

• Developed an invertible map between bipartite Bell scenarios and contextuality scenarios.
• Analyzed the transformation of correlation types (local, quantum, no-signaling) under this map.
• Utilized results from computational complexity theory, specifically MIP*=RE.

Main Results:

• Local, quantum, and no-signaling correlations map to noncontextual, quantum, and contextual behaviors, respectively.
• The set of quantum contextual behaviors is undecidable.
• Quantum contextual behaviors cannot be fully realized by finite-dimensional quantum systems and the set is not closed.
• This set and its closure are not limits of computable supersets.

Conclusions:

• The map provides a powerful tool for translating problems between Bell nonlocality and contextuality.
• The undecidability and non-closure results highlight fundamental limitations in characterizing quantum contextuality.
• The findings have implications for the study of quantum computational complexity and the nature of quantum reality.