  Symmetric extensions are essential in quantum mechanics, providing a lens to
investigate the correlations of entangled quantum systems and to address
challenges like the quantum marginal problem. Though semi-definite programming
(SDP) is a recognized method for handling symmetric extensions, it grapples
with computational constraints, especially due to the large real parameters in
generalized qudit systems. In this study, we introduce an approach that adeptly
leverages permutation symmetry. By fine-tuning the SDP problem for detecting \(
k \)-symmetric extensions, our method markedly diminishes the searching space
dimensionality and trims the number of parameters essential for positive
definiteness tests. This leads to an algorithmic enhancement, reducing the
complexity from \( O(d^{2k}) \) to \( O(k^{d^2}) \) in the qudit \( k
\)-symmetric extension scenario. Additionally, our approach streamlines the
process of verifying the positive definiteness of the results. These
advancements pave the way for deeper insights into quantum correlations,
highlighting potential avenues for refined research and innovations in quantum
information theory.
