Sufficient criteria for absolute separability in arbitrary dimensions via linear map inverses

Quantum states that remain separable (i.e. not entangled) under any global unitary transformation are known as<i>absolutely separable</i>and form a convex set. Despite extensive efforts, the complete characterization of this set remains largely unknown. In this work, we employ linear maps and their inverses to derive new sufficient analytical conditions for absolute separability in arbitrary dimensions, providing extremal points of this set and improving its characterization. Additionally, we employ convex geometry optimization to refine the characterization of the set when multiple non-comparable criteria for absolute separability are available. We also address the closely related problem of characterizing the<i>absolute PPT</i>(positive partial transposition) set, which consists of quantum states that remain positive under partial transposition across all unitary transformations. Finally, we extend our results to multipartite states.<i>We are proud to dedicate our work to Professor Ryszard Horodecki, whose pioneering contributions to the field of quantum entanglement continue to inspire us all. With deep gratitude and respect.</i>