Bounding conditional entropy of bipartite states with Bell operators

Quantum information theory explores numerous properties that surpass classical paradigms, offering novel applications and benefits. Among these properties, negative conditional von Neumann entropy (CVNE) is particularly significant in entangled quantum systems, serving as an indicator of potential advantages in various information-theoretic tasks, despite its indirect observability. In this paper, we investigate the relationship between CVNE and the violation of Bell inequalities. Our goal is to establish upper bounds on CVNE through semidefinite programming applied to entangled qubits and qutrits, utilizing selected Bell operators. Our findings reveal that a semi-device-independent certification of negative CVNE is achievable and could be practically beneficial. We further explore two types of robustness: robustness against detection efficiency loopholes, measured by relative violation, and robustness against white noise and imperfections in state preparation, measured by critical visibility. Additionally, we analyze parametrized families of Bell inequalities to identify optimal parameters for different robustness criteria. This study demonstrates that different Bell inequalities exhibit varying degrees of robustness depending on the desired properties, such as the type of noise resistance or the target level of negative CVNE. By bridging the gap between Bell inequalities and CVNE, our research enhances understanding of the quantum properties of entangled systems and offers insights for practical quantum information processing tasks.