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.

Positive maps constructed from free products provide entanglement tests, but their strength relative to standard criteria is not fully characterized. This paper compares the Collins–Hayden–Nechita free maps $\Phi_\mu$ with the reduction and positive partial transpose (PPT) criteria when $\mu$ is compactly supported and freely infinitely divisible. The main theorem states that, for every input dimension $n\ge2$ and ancillary dimension $m\ge1$, positivity of $\Phi_\mu$ implies $K_{\mathrm{RED}}^{(n,m)}\subseteq K_\mu^{(n,m)}$, where each set consists of states passing the corresponding test. Thus these maps detect no state that passes the reduction criterion and, since PPT implies reduction, cannot detect PPT entanglement. The bound is exact and dimension-independent rather than asymptotic. The proof constructs a joint full-Fock-space realization of the map’s corner variables, matching their mixed free cumulants. Positivity of the map yields a certificate in its free Lévy data: there exists $q\ge0$ such that $\begin{pmatrix}\gamma-q&v^*/\sqrt n\\v/\sqrt n&X/n+qI\end{pmatrix}\succeq0$. This step uses vacuum faithfulness, tensor-power test vectors, and an S-lemma argument. Normalizing a reduction-positive input gives a positive contraction; the certificate then lifts to an operator-valued block inequality ensuring positivity of the Fock realization. Moment matching and faithfulness transfer positivity back to the original map. The result is restricted to compactly supported freely infinitely divisible distributions. It does not resolve the broader comparison with PPT, or characterize states passing all positive free maps associated with unrestricted distributions.