A variant of the Kochen-Specker theorem localising value indefiniteness

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 KochenSpecker sets.If a system is prepared in a pure state |ψ , 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 |ψ , 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.

Experimentally probing the algorithmic randomness and incomputability of quantum randomness

“…Since the state |0 is thus an eigenstate of the projection observable P 0 = |0 0 |, this observable is value definite with value v(P 0 ) = 0-that is, the measurement outcome a 0 never occurs. 2 However, by the results of [10,44], both P 1 = |1 1 | and P 2 = |2 2 | are value indefinite and, moreover, both outcomes a 1 and a 2 occur with probability 1/2 according to the Born rule (1). Thus, the QRNG operates as an ideal coin certified by value indefiniteness.…”

Understanding quantum mechanics: a review and synthesis in precise language

“…In this case, the Bell-Kochen-Specker theorem does not imply that the results depend on the context [371]. -A third approach suggests that systems may not have definite values for all magnitudes, and so measurement event outcomes might not be determined in advance: a value-indefinite independent reality [372,373]. In other words, systems may not have case-properties (determinately valued properties) for all their type-properties (determinable properties) [374], a radical revision to more common views [375].…”

A Note on the Statistical Sampling Aspect of Delayed Choice Entanglement Swapping

“…Already at this point, it could quite justifiably be objected that this is an improper model for quantum singlets, as it implies that the two particles constituting the singlet have definite individual observable values. In contrast, a singlet quantum state is solely defined in terms of the correlations (joint probability distributions) [8][9][10], or, by another term, the relational properties [11,12] among the quanta; whereby (with some reasonable side assumptions such as non-contextuality) the supposition that the quanta carry additional information about their definite individual states leads to a complete contradiction [13,14].…”

“…In answering this question one can consult another paper by Peres [7] on the hypothetical (non-)existence of counterfactuals (or, in Specker's scholastic terminology [15], Infuturabilien). One of the most striking differences is the fact that classical configurations allow a truth table (that is, physical properties) of the constituents of the singlets, whereas hypothetical (counterfactual) truth tables associated with entangled quantum states, when viewed at different directions or contexts, in general do not; at least not statistically [7], but also not on a per particle pair basis [13,14].…”