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

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## Abstract

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 *in*deterministic 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.

## Keywords

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

### Acknowledgments

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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