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Concept

Measurement

In quantum mechanics, a complete set of yes-or-no questions, one for each direction of an orthonormal basis, of which exactly one is answered yes. The state fixes the probabilities of the answers, and the Kochen–Specker theorem says the answers themselves cannot all be fixed in advance.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

One threshold narrowing, one staying wide. Probability curves for connectivity and for containing a triangle, plotted against the edge probability as a multiple of each property's own threshold, at several graph sizes.

Sharp, or merely a threshold

Every monotone property of a random graph has a threshold. Some of them turn on over a range that shrinks relative to the threshold as the graph grows, and some do not — and which kind a property is turns out to be decided by whether it is about a local structure or about the whole graph.

probability · Random graphs Nine bases that cannot each have one true direction. Eighteen vectors (0, 0, 0, 1) (0, 0, 1, 0) (1, 1, 0, 0) (1, −1, 0, 0) (0, 1, 0, 0) (1, 0, 1, 0) (1, 0, −1, 0) (1, −1, 1, −1) (1, −1, −1, 1) (0, 0, 1, 1) (1, 1, 1, 1) (0, 1, 0, −1) (1, 0, 0, 1) (1, 0, 0, −1) (0, 1, −1, 0) (1, 1, −1, 1) (1, 1, 1, −1) (1, −1, −1, −1); vector i joins bases 1–2, 1–5, 1–3, 1–7, 2–5, 2–8, 2–4, 3–4, 3–6, 3–7, 4–6, 4–8, 5–9, 5–6, 6–9, 7–8, 7–9, 8–9; near-solution true: 2, 6, 10, 11, basis B9 empty.

No way to hand out true and false

Birkhoff and von Neumann read the propositions of quantum mechanics as subspaces, with "and" the intersection and "or" the span. Three lines in a plane already break the distributive law. Kochen and Specker proved something stronger: the propositions cannot be called true and false at all, if every complete measurement is to have exactly one true outcome. Cabello's eighteen directions in four dimensions show why in one line of arithmetic — nine bases, each direction in two of them, and nine is odd — and a search of all 262,144 assignments confirms it.

logic · Non classical logic

Named alongside it

The objects these essays reach for when they reach for this one.

Exhaustive searchImpossibilityLatticeLocalityMany valued logicMonotone propertyOrthogonalityParityPerfect matchingPhase transitionRandom graphSubspace

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