Foundations of Physics

, Volume 44, Issue 11, pp 1125–1155 | Cite as

The Status of Determinism in Proofs of the Impossibility of a Noncontextual Model of Quantum Theory

  • Robert W. SpekkensEmail author


In order to claim that one has experimentally tested whether a noncontextual ontological model could underlie certain measurement statistics in quantum theory, it is necessary to have a notion of noncontextuality that applies to unsharp measurements, i.e., those that can only be represented by positive operator-valued measures rather than projection-valued measures. This is because any realistic measurement necessarily has some nonvanishing amount of noise and therefore never achieves the ideal of sharpness. Assuming a generalized notion of noncontextuality that applies to arbitrary experimental procedures, it is shown that the outcome of a measurement depends deterministically on the ontic state of the system being measured if and only if the measurement is sharp. Hence for every unsharp measurement, its outcome necessarily has an indeterministic dependence on the ontic state. We defend this proposal against alternatives. In particular, we demonstrate why considerations parallel to Fine’s theorem do not challenge this conclusion.


Quantum contextuality Kochen–Specker theorem Quantum foundations Positive operator valued measures Quantum measurement theory 



The author would like to thank John Sipe and Howard Wiseman for their insistence that the topic of outcome-determinism for unsharp measurements deserved a better treatment. Thanks also to Ernesto Galvão, Ben Toner, Howard Wiseman, and Ravi Kunjwal for discussions, and to an anonymous referee for suggesting a simplification of the proof of Theorem 1. Research at Perimeter Institute is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MRI.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Perimeter Institute for Theoretical PhysicsWaterlooCanada

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