[Paper Review] Value Indefiniteness Is Almost Everywhere

AA Abbott, CS Calude+1 more|arXiv (Cornell University)|Sep 27, 2013

Quantum Mechanics and Applications4 citations

TL;DR

This paper demonstrates that value indefiniteness in quantum mechanics is not limited to isolated cases but occurs almost everywhere in the Hilbert space, under the assumption of local non-contextuality. By fixing a single observable as value definite, the theory implies that all other observables are value indefinite with Lebesgue measure one, establishing quantum indeterminacy as a global property.

ABSTRACT

Kochen-Specker theorems assure the breakdown of certain types of non-contextual hidden variable theories through the non-existence of global, holistic frame functions; alas they do not allow us to identify where this breakdown occurs, nor the extent of it. Here we show that this breakdown is maximal in that it occurs almost everywhere, and thus prove that quantum indeterminacy—often referred to as contextuality or value indefiniteness—is a global property as is often assumed. In contrast to the Kochen-Specker theorem, we only assume the weaker non-contextuality condition that any potential value assignments that may exist are locally non-contextual. Under this assumption, we prove that once a single arbitrary observable is fixed to be value definite, almost (i.e. with Lebesgue measure one) all remaining observables are indeterminate.

Motivation & Objective

• To investigate the extent and distribution of value indefiniteness in quantum systems beyond the scope of the Kochen-Specker theorem.
• To determine whether value indefiniteness is localized or pervasive across the space of observables.
• To establish that fixing a single observable as value definite leads to almost all others being value indefinite, under local non-contextuality.
• To clarify the global nature of quantum indeterminacy, challenging the assumption that contextuality is confined to specific configurations.

Proposed method

• Assume local non-contextuality: any value assignment to observables is consistent within local contexts but not necessarily globally.
• Fix one arbitrary observable to be value definite, using a frame function to represent its definite value assignment.
• Apply measure-theoretic analysis on the Hilbert space to evaluate the Lebesgue measure of the set of value-definite observables.
• Use the non-existence of global frame functions (as in Kochen-Specker) as a foundation to derive the measure-theoretic consequence of value indefiniteness.
• Demonstrate that under the local non-contextuality assumption, the set of value-definite observables has measure zero, implying almost all observables are value indefinite.
• Leverage the structure of quantum logic and orthomodular lattices to formalize the transition from local to global consistency.

Experimental results

Research questions

• RQ1 How widespread is value indefiniteness in quantum systems when only local non-contextuality is assumed?
• RQ2 What is the measure-theoretic extent of value-definite observables in the Hilbert space after fixing one observable as value definite?
• RQ3 Does the breakdown of non-contextual hidden variables occur only in isolated cases or is it pervasive?
• RQ4 Can the global nature of quantum indeterminacy be rigorously established under weaker assumptions than those in the Kochen-Specker theorem?
• RQ5 Is value indefiniteness a generic feature of quantum mechanics rather than a rare exception?

Key findings

• Once a single observable is fixed to be value definite, the set of all other observables that can be consistently assigned definite values has Lebesgue measure zero.
• The remaining observables—almost all of them—are value indefinite, meaning value indefiniteness is generic in the quantum state space.
• The result holds under the assumption of local non-contextuality, which is weaker than the global non-contextuality assumed in the Kochen-Specker theorem.
• The breakdown of non-contextual hidden variable theories is not confined to specific configurations but occurs almost everywhere in the Hilbert space.
• This confirms that quantum indeterminacy is a global property, consistent with the widespread belief in contextuality as a fundamental feature of quantum mechanics.
• The measure-theoretic conclusion implies that value-definite assignments are exceptional, not typical, in the space of all possible quantum observables.