Logarithmic negativity in the qubit–qudit case

Determine the relationship between the logarithmic negativity E_N(ρ) and the zero-error PPT entanglement cost E_{c,PPT}^{exact}(ρ) for qubit–qudit systems (i.e., bipartite systems with minimal local dimension d = 2 but |B| > 2). Specifically, establish whether E_{c,PPT}^{exact}(ρ) = E_N(ρ) for all such states or characterize the class of qubit–qudit states for which equality holds.

Background

For two-qubit states it is known (Ishizaka) that the zero-error PPT entanglement cost equals the logarithmic negativity, since all two-qubit states have zero bi-negativity. The authors proved that for qubit–qudit systems E_{c,PPT}^{exact} = E_χ = E_κ, but a closed-form expression via E_N is not established.

Understanding whether E_N suffices in the qubit–qudit case would yield a closed-form single-letter formula for the entanglement cost and generalize known two-qubit results.