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Quantum complementarity and logical indeterminacy

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Abstract

Whenever a mathematical proposition to be proved requires more information than it is contained in an axiomatic system, it can neither be proved nor disproved, i.e. it is undecidable, or logically undetermined, within this axiomatic system. I will show that certain mathematical propositions on a d-valent function of a binary argument can be encoded in d-dimensional quantum states of mutually unbiased basis (MUB) sets, and truth values of the propositions can be tested in MUB measurements. I will then show that a proposition is undecidable within the system of axioms encoded in the state, if and only if the measurement associated with the proposition gives completely random outcomes.

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Notes

  1. For a possible relation between Gödel’s theorem and physics see Barrow (2006). For a relation between algorithmic randomness and quantum indeterminacy see Svozil (1990), Stay and Calude (2005) and Calude and Svozil (2006).

  2. The considerations here can be generalized to all dimensions that are powers of primes. This is related to the fact that in quantum theory in these cases a complete set of mutually unbiased bases is known to exit. In all other cases this is an open question and goes beyond the scope of this paper (see, for example Paterek et al. (2009)).

  3. To put it in a grotesque way the system is not allowed to response "I am undecidable, I cannot give an answer."

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Authors and Affiliations

  1. Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, 1090, Vienna, Austria

    Časlav Brukner

  2. Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090, Vienna, Austria

    Časlav Brukner

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  1. Časlav Brukner
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Correspondence to Časlav Brukner.

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Brukner, Č. Quantum complementarity and logical indeterminacy. Nat Comput 8, 449–453 (2009). https://doi.org/10.1007/s11047-009-9118-z

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