Generalizations of Kochen and Specker's theorem and the effectiveness of Gleason's theorem

Abstract

Abstract. Kochen and Specker's theorem can be seen as a consequence of Gleason's theorem and logical compactness. Similar compactness arguments lead to stronger results about finite sets of rays in Hilbert space, which we also prove by a direct construction. Finally, we demonstrate that Gleason's theorem itself has a constructive proof, based on a generic, finite, effectively generated set of rays, on which every quantum state can be approximated.

What Is So Special about Quantum Clicks?

“…“Extensions” of the Kochen-Specker theorem investigate situations in which a system is prepared in a state along direction and measured along a non-orthogonal, non-collinear projection along direction . Those extensions yield what may be called [ 120 , 200 ] indeterminacy : Pitowsky’s logical indeterminacy principle (Theorem 6, p. 226 in [ 120 ]) states that given two linearly independent non-orthogonal unit vectors and in , there is a finite set of unit vectors containing and for which the following statements hold: There is no two-valued state v on which satisfies either , or and , or and . …”

New Forms of Quantum Value Indefiniteness Suggest That Incompatible Views on Contexts Are Epistemic

“…Probably the strongest forms of value indefiniteness [ 61 , 62 ] are theorems [ 51 , 63 , 64 ] stating that relative to reasonable (admissibility, non-contextuality) assumptions, if a quantized system is prepared in some pure state , then any observable that is not identical or orthogonal to is undefined. That is, there exist finite systems of quantum contexts whose pastings are demanding that any pure state not belonging to some context with can neither be true, nor false; else a complete contradiction would follow from the assumption of classically pre-existent truth values on some pasting of contexts such as the Specker bug logic.…”

Unscrambling the Quantum Omelette

“…A formalism defining partial frame functions, similar to the one developed in Ref. [27,28] (instead of the "holistic" frame function defined everywhere by Pitowsky's logical indeterminacy principle [29,30]) can, in a particular sense, be considered an "improved" version of the KS theorem which certifies "breakdown of (non-contextual) value definiteness" for any observable |b b| (associated with the vector |b ; from now on, the vector and its associated projector will be used synonymously), if the quantum is prepared in a particular state such that the observable |c , which must be non-orthogonal and noncollinear to |b , occurs with certainty. More formally, by considering some finite construction of interconnected contexts Γ(C 1 , C 2 , .…”