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This study derives quantum probabilities from measurement context, not as an assumption. It quantifies contextuality using information geometry, unifying quantum probability and classical structure.

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Area of Science:

  • Quantum Information Theory
  • Foundations of Quantum Mechanics
  • Mathematical Physics

Background:

  • The Born rule, fundamental to quantum mechanics, provides probabilities for measurement outcomes.
  • Existing approaches often assume the Born rule or rely on complex frameworks.
  • Understanding the origin and context-dependence of quantum probabilities is crucial.

Purpose of the Study:

  • To provide a variational characterization of the Born rule.
  • To quantify quantum contextuality using information-geometric principles.
  • To unify quantum probability, contextuality, and classical structures via optimization.

Main Methods:

  • Projecting quantum states onto abelian algebras by minimizing Umegaki relative entropy.
  • Utilizing Petz's Pythagorean identity to derive Born weights.
  • Measuring contextuality via classical Kullback-Leibler distance to a noncontextual polytope.
  • Employing sheaf theory and categorical structures.

Main Results:

  • Born weights are derived as a consequence of minimizing relative entropy, not assumed.
  • A convex objective function Φ(ρ) quantifies contextuality, equaling zero for noncontextual theories.
  • The closest noncontextual model is identified as the classical I-projection.
  • The framework extends to various quantum measurement formalisms (PVMs, POVMs).

Conclusions:

  • This work offers a novel, information-geometric derivation of the Born rule.
  • It explicitly addresses and quantifies quantum contextuality and the relational nature of probabilities.
  • The approach unifies diverse concepts into a single optimization principle for quantum foundations.