Entanglement measure for general pure two-qudit concentration

Identify a suitable entanglement measure that characterizes the parallel entanglement-concentration rule for the normalized concentration map on arbitrary pure two-qudit states in terms of the entanglement measures of the initial states.

Background

For pure two-qubit states, the paper expresses the output concurrence of the deterministic concentration protocol in terms of the input concurrences. It then considers the more general case of pure two-qudit states, where the concentration output is described through its Schmidt coefficients using majorization theory.

The paper does not provide an entanglement measure yielding an analogous expression for the output based on the entanglement of the two initial qudit states. Finding such a measure would extend the operator-percolation framework beyond the qubit setting.

References

However, to date, finding a suitable entanglement measure that characterizes the parallel rule for $\Psi_{ST}$ in terms of the entanglement of the initial states remains an open problem.

— An Operator-Theoretic Model of Entanglement Percolation in Series-Parallel Quantum Networks  (2609.26367 - He et al., 22 Sep 2026) in Example 4.3, Section 4, after Eq. (\ref{eq-qudit_concen})

However, whether these properties hold for general entangled states and entanglement measures remains an open question.

— An Operator-Theoretic Model of Entanglement Percolation in Series-Parallel Quantum Networks  (2609.26367 - He et al., 22 Sep 2026) in Example 4.4, Section 4, following Eq. (\ref{eq-para_chi})