A partial-trace matrix inequality and Werner-state distillability

Abstract

Motivated by the equivalent partial-trace formulations of Werner-state distillability [P. Costa Rico, Lett. Math. Phys. 115, 47 (2025); S.-Y. Qi et al., Phys. Rev. A 110, 012406 (2024)], we prove a bipartite partial-trace inequality for every matrix of rank at most two. As applications, we prove the two-copy undistillability of NPT Werner states in arbitrary local dimension, thereby resolving this open problem highlighted in [P. Horodecki et al., PRX Quantum 3, 010101 (2022)]. We further prove a two-parameter extension of the matrix inequality and show that two individually one-copy-undistillable NPT Werner states cannot activate each other's one-copy distillability. We also resolve the singular-value maximization problem associated with the two-ququart case.

arXiv comment9 pages, no figures. The proof of the main result, namely Theorem 5, was completed in June 2026, and the remaining results and manuscript were finalized in the following month