Collapse of the χ-hierarchy at a finite level

Determine whether the χ-hierarchy of PPT entanglement monotones collapses at a finite level for some or all bipartite quantum states. Specifically, establish whether there exists an integer p* such that for every bipartite state ρ the equality E_{χ,p}(ρ) = E_{χ,p*}(ρ) holds for all p ≥ p*, where E_{χ,p}(ρ) = log₂ χ_p(ρ) and χ_p(ρ) is defined by the semidefinite program that minimizes Tr S_p subject to −S_i ≤ S_{i−1}^Γ ≤ S_i for i = 0,…,p with S_{−1} = ρ.

Background

The paper introduces two hierarchies of semidefinite programs, the χ-hierarchy and the κ-hierarchy, that bound and converge to the zero-error PPT entanglement cost for any bipartite state. The authors prove exponential convergence and provide an efficient algorithm to compute the cost without a closed-form expression.

A collapse of the χ-hierarchy would imply that only a finite level p* is needed to exactly compute the entanglement cost for all states, yielding a simple single-letter formula. This would further simplify both analysis and computation beyond the already efficient hierarchical approach.