# Robust self-testing of two-qubit states

## TL;DR

This paper investigates the robustness of self-testing two-qubit states via tilted CHSH inequalities, providing numerical evidence for bounds and establishing a threshold for CHSH violation to qualify as a self-test.

## Contribution

It introduces a new bound for robust self-testing using tilted CHSH inequalities and proves a threshold condition for CHSH violation to serve as a self-test.

## Key findings

- Numerical evidence supports the proposed bound for tilted CHSH inequalities.
- A proof shows CHSH inequality requires violation above a threshold to be a self-test.
- Self-testing scenarios are classified into two types based on the presence of a violation threshold.

## Abstract

It is well-known that observing nonlocal correlations allows us to draw conclusions about the quantum systems under consideration. In some cases this yields a characterisation which is essentially complete, a phenomenon known as self-testing. Self-testing becomes particularly interesting if we can make the statement robust, so that it can be applied to a real experimental setup. For the simplest self-testing scenarios the most robust bounds come from the method based on operator inequalities. In this work we elaborate on this idea and apply it to the family of tilted CHSH inequalities. These inequalities are maximally violated by partially entangled two-qubit states and our goal is to estimate the quality of the state based only on the observed violation. For these inequalities we have reached a candidate bound and while we have not been able to prove it analytically, we have gathered convincing numerical evidence that it holds. Our final contribution is a proof that in the usual formulation, the CHSH inequality only becomes a self-test when the violation exceeds a certain threshold. This shows that self-testing scenarios fall into two distinct classes depending on whether they exhibit such a threshold or not.