  This paper presents a comprehensive exploration of semi-definite programming
(SDP) techniques within the context of quantum information. It examines the
mathematical foundations of convex optimization, duality, and SDP formulations,
providing a solid theoretical framework for addressing optimization challenges
in quantum systems. By leveraging these tools, researchers and practitioners
can characterize classical and quantum correlations, optimize quantum states,
and design efficient quantum algorithms and protocols. The paper also discusses
implementational aspects, such as solvers for SDP and modeling tools, enabling
the effective employment of optimization techniques in quantum information
processing. The insights and methodologies presented in this paper have proven
instrumental in advancing the field of quantum information, facilitating the
development of novel communication protocols, self-testing methods, and a
deeper understanding of quantum entanglement. Overall, this study offers a
resource for researchers interested in the intersection of optimization and
quantum information, opening up new avenues for exploration and breakthroughs
in this rapidly evolving field.
