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. 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? → Irreducible Randomness Zeilinger, 1999 Brukner, Zeilinger 2002 quant-ph/0212084

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

4. Closer look … 1 bit available → 3 logically complementary statements 12 3

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

6. How to represent mathematical functions physically?

7. Testing theorems using quantum mechanics

8. Testing Undecidable Theorems State PreparationMeasurement Bases y z x y z x 1.Systems give answers when asked (detectors “click”) 2.The question asked is undecidable → Random results! Is quantum randomness physical expression for mathematical undecidability?