Existence of pure states with maximally mixed two-particle reduced density matrices

Determine for which pairs of particle number and mode number 8M,N9 a pure fermionic state can realize a maximally mixed two-particle reduced density matrix and thereby attain the universal minimum of the connected quartic norm.

Background

The paper shows that, on the manifold of pure fixed-particle-number states with flat one-body reduced density matrix, the Frobenius norm of the connected two-body reduced density-matrix cumulant is equivalent to the purity of the normalized 2RDM. A universal lower bound follows from the trace of this cumulant, and equality requires the normalized 2RDM to be maximally mixed on the antisymmetric two-particle space.

The authors identify the attainability of this lower bound as a pure-state representability problem: a maximally mixed 2RDM automatically implies a flat 1RDM, but it is not known for which 8M,N9 such a 2RDM can arise from a pure N-fermion state.