Spectral bounds for the partial transpose

Abstract:Among the various entanglement measures, the negativity stands out not only for its clear physical meaning but also for being directly computable from the spectrum of the partial transpose. However, the negativity captures only the total weight of the negative eigenvalues, whereas the finer structure of the negative spectrum remains largely unexplored. In this work, we fill this gap by introducing a hierarchical generalization of the negativity, the Ky Fan $k$-negativity, and developing a unified analytical framework for deriving spectral bounds on the partial transpose. To obtain the absolute bounds, we reduce the maximization problem to a spectral graph optimization, whose solution yields the exact bound for every $k$ through a single cubic equation. We further investigate these spectral bounds both analytically and numerically under a fixed-purity constraint. In particular, we solve the $k=1$ case completely and uncover a simple underlying graph structure. As two direct applications, we show that the Ky Fan $k$-negativity robustly certifies genuine multilevel entanglement and that it converts the $p_3$-PPT condition into a quantitative lower bound on the negativity.

arXiv:2609.06617 [quant-ph]

(or arXiv:2609.06617v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.06617 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.