A Reduction Bound for Entanglement Detection by Positive Free Maps

Abstract

Collins, Hayden, and Nechita used free probability to construct families of $k$-positive but not $(k+1)$-positive maps and applied them to entanglement detection. We prove that positive free maps associated with compactly supported freely infinitely divisible distributions are no stronger than the reduction criterion in every ancillary dimension. In particular, these maps cannot detect positive partial transpose entanglement. The proof combines a joint Fock realization with an operator positivity certificate derived from free Lévy data.