# Analytic Semi-device-independent Entanglement Quantification for Bipartite Quantum States

## TL;DR

This paper introduces a semi-device-independent method to analytically quantify entanglement in bipartite quantum states using nondegenerate Bell inequalities, applicable with only measurement statistics.

## Contribution

It defines nondegeneracy for Bell inequalities and proves that many known inequalities are nondegenerate, enabling entanglement bounds from measurement data.

## Key findings

- Provides an analytic lower bound for entanglement of formation and distillable entanglement.
- Characterizes the structure of nondegenerate Bell inequalities.
- Demonstrates the method on qutrit-qutrit states with CGLMP inequality.

## Abstract

We define a property called nondegeneracy for Bell inequalities, which describes the situation that in a Bell setting, if a Bell inequality and involved local measurements are chosen and fixed, any quantum state with a given dimension and its orthogonal quantum state cannot violate the inequality remarkably at the same time. By choosing a proper nondegenerate Bell inequality, we prove that for a unknown bipartite quantum state of a given dimension, based on the measurement statistics only, we can provide an analytic lower bound for the entanglement of formation or even for the distillable entanglement, making the whole process semi-device-independent. We characterize the mathematical structure of nondegenerate Bell inequalities, and prove that quite a lot of well-known Bell inequalities are nondegenerate. We demonstrate our approach by quantifying entanglement for qutrit-qutrit states based on their violation to the CGLMP inequality.