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Information-Theoretical Formulation of Gödel’s Theorem and Quantum Physics

Explore the connection between Gödel's theorem and quantum physics, showcasing irreducible randomness, undecidability, and experimental tests of theorems using quantum mechanics.

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Information-Theoretical Formulation of Gödel’s Theorem and Quantum Physics

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  1. Institute for Experimental Physics University of Vienna Institute for Quantum Optics and Quantum Information Austrian Academy of Sciences Mathematical Undecidability & Quantum Complementarity Časlav Brukner (in collaboration with Tomasz Paterek) Reykjavik, Iceland July 2007

  2. Information Information-theoretical formulation of quantum physics: The most elementary system contains one bit of information. → Irreducible Randomness Zeilinger, 1999 Brukner, Zeilinger 2002 quant-ph/0212084 Information-theoretical formulation of Gödel’s theorem: If a theorem contains more information than a given set of axioms, then it is impossible for the theorem to be derived from the axioms. Chaitin, 1982 Are the two related to each other?

  3. Undecidability: Simple Example Axiom: “f(0)= 0” 1bit Can be neither proved nor disproved: needs 2 bits Theorem: “f(0)=0 andf(1)=0” Theorem: “f(0)=0 andf(0) = f(1)”. Independent Statements: “f(0)= 0”, “f(1)= 0”, “f(0)=f(1)”

  4. Closer look … 1 bit available → 3 logically complementary statements 3 1 2

  5. 2 2 3 3 1 1 Axiom Theorem „Experimental Test of Theorems“ 2 3 1 2 3 1 Axiom Theorem

  6. How to represent mathematical functions physically?

  7. Testing theorems using quantum mechanics

  8. Testing Undecidable Theorems y y z x z x State Preparation Measurement Bases • Systems give answers when asked (detectors “click”) • The question asked is undecidable • → Random results! Is quantum randomness physical expression for mathematical undecidability?

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