Statistical signatures of quantum contextuality

TL;DR

This paper introduces a new method for reconstructing quantum states using the relations between measurement contexts that exhibit quantum contextuality, revealing a deterministic structure in measurement outcomes.

Contribution

It presents a novel reconstruction technique based on contextual relations, utilizing a set of eleven elements to describe quantum states in a three-dimensional Hilbert space.

Findings

A set of five relations links measurement contexts deterministically.

An overcomplete set of eleven elements provides an unbiased state description.

The method demonstrates consistent contextual descriptions across examples.

Abstract

Quantum contextuality describes situations where the statistics observed in different measurement contexts cannot be explained by a measurement independent reality of the system. The most simple case is observed in a three-dimensional Hilbert space, with five different measurement contexts related to each other by shared measurement outcomes. The quantum formalism defines the relations between these contexts in terms of well-defined relations between operators, and these relations can be used to reconstruct an unknown quantum state from a finite set of measurement results. Here, I introduce a reconstruction method based on the relations between the five measurement contexts that can violate the bounds of non-contextual statistics. A complete description of an arbitrary quantum state requires only five of the eight elements of a Kirkwood-Dirac quasi probability, but only an overcomplete set of eleven elements provides an unbiased description of all five contexts. A set of five fundamental relations between the eleven elements reveals a deterministic structure that links the five contexts. As illustrated by a number of examples, these relations provide a consistent description of contextual realities for the measurement outcomes of all five contexts.