Mathematical undecidability and quantum randomness
Tomasz Paterek, Časlav Brukner, Robert Prevedel +4 more
We propose a new link between mathematical undecidability and quantum physics. We demonstrate that the states of elementary quantum systems are capable of encoding mathematical axioms and show that quantum measurements are capable of revealing whether a given proposition is decidable or not within the axiomatic system. Whenever a mathematical proposition is undecidable within the axioms encoded in the state, the measurement associated with the proposition gives random outcomes. Our results support the view that quantum randomness is irreducible and a manifestation of mathematical undecidability. Whenever a proposition and a given set of axioms together contain more information than the axioms themselves, the proposition can neither be proved nor disproved from the axioms { it is mathematically undecidable [1, 2]. Here we propose a novel link between mathematical undecidability and quantum physics. We demonstrate that the states of elementary quantum systems are capable of encoding mathematical axioms. Quantum mechanics imposes an upper limit on how much information can be encoded in a quantum state [3, 4], thus limiting the information content of the set of axioms. We show that quantum measurements are capable of revealing whether a given proposition is decidable or not. Whenever a mathematical proposition is undecidable within the system of axioms encoded in the state, the measurement associated with the proposition gives random outcomes. This allows for an experimental test of mathematical undecidability by realizing in the laboratory both the actual quantum states and the required quantum measurements. (To illustrate these ideas, we conducted experiments using the polarization of photons.) Our results support the view that quantum randomness is irreducible [5, 6] and a manifestation of mathematical undecidability. Any formal system is based on axioms, which are propositions that are dened to be true. A proposition is logically independent from a given set of axioms if it can neither be proved nor disproved from the axioms. If a proposition is independent from the axioms, neither the proposition itself nor its negation creates an inconsistency together with the axiomatic system. ^
•Dec 1, 2008•1 citation