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Adapting Logic to Physics: The Quantum-Like Eigenlogic Program.

Zeno Toffano1, François Dubois2

  • 1Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des signaux et systèmes, 91190 Gif-sur-Yvette, France.

Entropy (Basel, Switzerland)
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Summary
This summary is machine-generated.

Eigenlogic, a new operator-based logic inspired by quantum theory, redefines logical propositions using linear algebra. This quantum logic approach extends classical systems, enabling fuzzy logic and new possibilities for quantum information technologies.

Keywords:
operator algebraprobabilistic logicquantum computing gates

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Area of Science:

  • * Quantum Information Science
  • * Mathematical Logic
  • * Theoretical Physics

Background:

  • * Rapid advancements in quantum information technologies necessitate novel logical frameworks.
  • * Classical logic, based on Boolean algebra, has limitations in representing complex quantum phenomena.
  • * Existing logical systems do not fully capture the nuances of quantum computation.

Purpose of the Study:

  • * To introduce Eigenlogic, a novel operator-based logical system inspired by quantum theory.
  • * To extend classical logic by employing linear algebra and operator methods.
  • * To explore the connection between quantum principles and logical structures for enhanced computational capabilities.

Main Methods:

  • * Representing logical propositions using linear algebra and operators.
  • * Utilizing eigenvalue structures to define logical truth tables.
  • * Employing projection operators for the binary alphabet {0, 1} and reversible involution operators for {+1, -1}.
  • * Synthesizing many-valued logical operators via Lagrange interpolation and the Cayley-Hamilton theorem.
  • * Applying quantum probability (Born rule) for fuzzy logic representation.

Main Results:

  • * Eigenlogic successfully expresses logical propositions through linear algebra operators.
  • * The system extends classical logic to non-binary alphabets and fuzzy logic representations.
  • * Operator methods enable the synthesis of diverse logical operators, including many-valued ones.
  • * The eigenvalue structure of Eigenlogic is linked to the universality of quantum gates, highlighting non-commutativity and entanglement.

Conclusions:

  • * Eigenlogic provides a powerful new framework for logical reasoning, deeply integrated with quantum principles.
  • * The operator-based approach offers a flexible and extensible alternative to classical logic.
  • * This research opens avenues for developing advanced quantum information processing and computation.