Tapping into Permutation Symmetry for Improved Detection of k-Symmetric Extensions

TL;DR

This paper presents a novel method leveraging permutation symmetry to significantly improve the efficiency of detecting k-symmetric extensions in quantum systems, reducing computational complexity and enhancing positive definiteness verification.

Contribution

The authors introduce a permutation symmetry-based approach that reduces the complexity of SDP problems for k-symmetric extensions from exponential to polynomial scale.

Findings

Reduces SDP complexity from O(d^{2k}) to O(k^{d^2})

Streamlines positive definiteness verification process

Enables deeper analysis of quantum correlations

Abstract

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.