Unrestricted global optimality of the Higuchi–Sudbery entropy criterion

Determine whether the four-qubit Higuchi–Sudbery state globally maximizes the average two-versus-two bipartite von Neumann entropy over all four-qubit pure states, rather than only within the 1-uniform family.

Background

The Higuchi–Sudbery state M4M_4 is shown in the paper to globally maximize the geometric measure of four-qubit entanglement and the average two-versus-two linear entropy. For the average von Neumann entropy originally considered by Higuchi and Sudbery, the state is a proven local maximum and is globally maximal within the family of four-qubit 1-uniform states.

The unrestricted optimization problem over all four-qubit pure states remains unresolved. The paper records this as a conjectural extremality claim rather than proving it.

References

Higuchi and Sudbery originally proposed it using the corresponding average von Neumann entropy; it is a proven local maximum and a global maximum within the 1-uniform family for that functional, while unrestricted global optimality remains conjectural .

— The Entanglement Content of Quantum Measurement Bases  (2608.21185 - Pauwels et al., 21 Aug 2026) in Section 4.2, paragraph beginning “The example is independently notable for its entanglement”