Abstract
The Kochen-Specker theorem is a fundamental result in quantum foundations that has spawned massive interest since its inception. We show that within every Kochen-Specker graph, there exist interesting subgraphs which we term 01-gadgets, that capture the essential contradiction necessary to prove the Kochen-Specker theorem, i.e,. every Kochen-Specker graph contains a 01-gadget and from every 01-gadget one can construct a proof of the Kochen-Specker theorem. Moreover, we show that the 01-gadgets form a fundamental primitive that can be used to formulate state-independent and state-dependent statistical Kochen-Specker arguments as well as to give simple constructive proofs of an ``extended'' Kochen-Specker theorem first considered by Pitowsky in \cite{Pitowsky}.
Highlights
According to the quantum formalism, a projective measurement M is described by a set M = {V1, . . . , Vm} of projectors Vi in a complex Hilbert space, that are orthogonal, ViVj = δijVi, and sum to the identity, i Vi = I
It turns out that the statistical proofs of the KochenSpecker theorem can be interpreted in the same manner
We have shown that there exist interesting subgraphs of the Kochen-Specker graphs that we termed 01-gadgets that encapsulate the main contradiction necessary to prove the Kochen-Specker theorem
Summary
According to the quantum formalism, a projective measurement M is described by a set M = {V1, . Though quantum measurements are defined by sets of projectors, the outcome probabilities of these measurements are determined by the individual projectors alone, independently of the broader set – or the context – to which they belong. The Kocken-Specker (KS) theorem [1] is a cornerstone result in the foundations of quantum mechanics, establishing that, in Hilbert spaces of dimension greater than two, it is not possible to find a deterministic outcome assignment that is non-contextual. Non-contextual means, as above, that these probabilities are not directly assigned to the measurements themselves, but to the individual projectors from which they are composed, independently of the context to which the projectors belong. The KS theorem establishes that it is not possible to find a rule f such that
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