The complexity of semidefinite programs for testing $k$-block-positivity

We extend \cite{chen2025srkbp} by analyzing the complexity of the $k$-block-positivity testing algorithm that stems from the optimization problem in Definition \ref{definition:SDP-k-block-positivity}. In this paper, we investigate a symmetry reduction scheme based on rectangular shaped Young diagrams. Connecting the complexity to the dimensions of irreducible representations of $\U(d)$, we derive an explicit formula for the complexity, which also clarifies why the semidefinite program hierarchy collapses in the $k=d$ case.

💡 Research Summary

This paper investigates the computational complexity of semidefinite programming (SDP) algorithms used to test k‑block‑positivity of operators—a central problem in quantum information theory related to the positivity of quantum channels, entanglement cost, and the duality with Schmidt‑rank‑k states. Building on the authors’ earlier work (referenced as