Unsharp eigenvalues and quantum contextuality

TL;DR

This paper investigates how assuming unsharp eigenvalues in quantum measurements affects the validity of tests for quantum contextuality and hidden-variable theories, revealing a new loophole in fundamental quantum tests.

Contribution

It demonstrates that unsharp eigenvalues prevent ruling out non-contextual hidden-variable theories, challenging the assumptions underlying key quantum foundational tests.

Findings

Unsharp eigenvalues invalidate the Kochen-Specker theorem assumptions

Quantum contextuality cannot be confirmed under unsharp eigenvalue assumptions

A new loophole in fundamental quantum tests is identified

Abstract

The Kochen-Specker theorem, Bell inequalities, and several other tests that were designed to rule out hidden-variable theories, assume the existence of observables having infinitely sharp eigenvalues. A paradigmatic example is spin-1/2. It is measured with a Stern-Gerlach array whose outputs are divided into two classes, spin-up and spin-down, in correspondence to the two spots observed on a detection screen. The spot's finite size is attributed to imperfections of the measuring device. This assumption turns the experimental output into a dichotomic, discrete one, thereby allowing the assignment of each spot to an infinitely sharp eigenvalue. Alternatively, one can assume that the spot's finite size stems from eigenvalues spanning a continuous range. Can we disprove such an assumption? Can we rule out hidden-variable theories that reproduce quantum predictions by assuming that, e.g., the electron's magnetic moment is not exactly the same for all electrons? We address these questions by focusing on the Peres-Mermin version of the Bell-Kochen-Specker theorem. It is shown that the assumption of unsharp eigenvalues precludes ruling out non-contextual hidden-variable theories and hence quantum contextuality does not arise. Analogous results hold for Bell-like inequalities. This represents a new loophole that spoils several fundamental tests of quantum mechanics and issues the challenge to close it.