T Paterek et al 2010 New J. Phys. 12 013019 doi:10.1088/1367-2630/12/1/013019
Logical independence and quantum randomness
T Paterek1,3,5, J Kofler1,2, R Prevedel2, P Klimek2,4, M Aspelmeyer1,2, A Zeilinger1,2 and Č Brukner1,2
Show affiliationsWe propose a link between logical independence and quantum physics. We demonstrate that quantum systems in the eigenstates of Pauli group operators are capable of encoding mathematical axioms and show that Pauli group quantum measurements are capable of revealing whether or not a given proposition is logically dependent on the axiomatic system. Whenever a mathematical proposition is logically independent of the axioms encoded in the measured state, the measurement associated with the proposition gives random outcomes. This allows for an experimental test of logical independence. Conversely, it also allows for an explanation of the probabilities of random outcomes observed in Pauli group measurements from logical independence without invoking quantum theory. The axiomatic systems we study can be completed and are therefore not subject to Gödel's incompleteness theorem.
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GENERAL SCIENTIFIC SUMMARY
Introduction and background. Quantum theory makes only probabilistic statements about individual measurement outcomes. In general, the outcomes have an element of randomness, the roots of which are still not fully understood. We propose a link between quantum randomness and logical independence. A proposition is logically independent from a set of axioms if it can neither be proved nor disproved from the axioms. A logically independent proposition represents entirely new information, which is not contained in the axioms.Main results. We show that quantum states from a certain class encode mathematical axioms and that corresponding measurements test the truth-values of mathematical propositions. Quantum mechanics imposes an upper limit on how much information can be carried by a quantum state ('N qubits carry N bits of information'), thus limiting the information content of the set of axioms. We show that whenever a mathematical proposition is logically independent of the axioms encoded in the state, the measurement associated to the proposition gives random outcomes. Whenever the proposition is logically dependent on the axioms, the measurement outcome is definite. This shines new light on the nature of quantum randomness without invoking the quantum formalism itself.
Wider implications. While we have studied only quantum mechanics, our reasoning applies to any theory of systems with limited information content in which physical states can encode mathematical propositions and measurements can be identified with questions about their truth values. Whenever the information required to assign the truth values of the propositions which are tested on the system exceeds its information content, measurement outcomes are inevitably random.
- PACS
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03.65.Ta Foundations of quantum mechanics; measurement theory
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
- Subjects
- Dates
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Issue 1 (January 2010)
Received 14 July 2009
Published 20 January 2010
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Logical independence and quantum randomness
T Paterek et al 2010 New J. Phys. 12 013019
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