### Abstract

The Kochen-Specker theorem proves the inability to assign, simultaneously, noncontextual definite values to all (of a finite set of) quantum mechanical observables in a consistent manner. If one assumes that any definite values behave noncontextually, one can nonetheless only conclude that some observables (in this set) are value indefinite. In this paper, we prove a variant of the Kochen-Specker theorem showing that, under the same assumption of noncontextuality, if a single one-dimensional projection observable is assigned the definite value 1, then no one-dimensional projection observable that is incompatible (i.e., non-commuting) with this one can be assigned consistently a definite value. Unlike standard proofs of the Kochen-Specker theorem, in order to localise and show the extent of value indefiniteness, this result requires a constructive method of reduction between Kochen-Specker sets. If a system is prepared in a pure state
, then it is reasonable to assume that any value assignment (i.e., hidden variable model) for this system assigns the value 1 to the observable projecting onto the one-dimensional linear subspace spanned by
, and the value 0 to those projecting onto linear subspaces orthogonal to it. Our result can be interpreted, under this assumption, as showing that the outcome of a measurement of any other incompatible one-dimensional projection observable cannot be determined in advance, thus formalising a notion of quantum randomness.

Received 09 March 2015
Accepted 11 September 2015
Published online 01 October 2015

Acknowledgments:
We thank the anonymous referees for suggestions which helped improve this paper. This work was supported in part by Marie Curie No. FP7-PEOPLE-2010-IRSES Grant RANPHYS.

Article outline:

I. THE KOCHEN-SPECKER THEOREM AND VALUE INDEFINITENESS
A. Definitions
B. The Kochen-Specker theorem
II. A PATH TO LOCALISING VALUE INDEFINITENESS
A. Localising the hypotheses
1. An illustrated example
III. THE LOCALISED VARIANT OF THE KOCHEN-SPECKER THEOREM
A. Insufficiency of existing Kochen-Specker diagrams
B. Proof of Theorem 2
IV. DISCUSSION
A. Proof size
B. State-independence and testability
V. CONCLUSIONS

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