# Geometry of joint reality: device-independent steering and operational completeness

## TL;DR

This paper explores the geometric and operational principles underlying quantum steering and Bell inequalities, introducing new device-independent criteria and demonstrating their implications for quantum nonlocality and contextuality.

## Contribution

It presents novel geometric and operational frameworks for understanding quantum steering and Bell inequalities, including weaker assumptions and finite statistics considerations.

## Key findings

- Derived device-independent steering inequalities based on geometry and locality.
- Established a Bell inequality from locality and one-sided reality assumptions.
- Extended Pusey's result to device-independent steering and operational completeness scenarios.

## Abstract

We look at what type of arguments can rule out the joint reality (or value definiteness) of two observables of a physical system, such as a qubit, and give several strong yet simple no-go results based on assumptions typically weaker than those considered previously. The first result uses simple geometry combined with a locality assumption to derive device-independent steering inequalities. These may also be regarded as "conditional" Bell inequalities, are simpler in principle to test than standard Bell inequalities, and for two-qubit systems are related to properties of the quantum steering ellipsoid. We also derive a Bell inequality from locality and a one-sided reality assumption, and demonstrate a close connection between device-independent steering and Bell nonlocality. Moreover, we obtain a no-go result without the use of locality or noncontextuality assumptions, based on similar geometry and an assumption that we call "operational completeness". The latter is related to, but strictly weaker than, preparation noncontextuality. All arguments are given for finite statistics, without requiring any assumption that joint relative frequencies converge to some (unobservable) joint probability distribution. We also generalise a recent strong result of Pusey, for preparation noncontextuality, to the scenarios of device-independent steering and operational completeness.