Closed-form existence criterion for golden states under FO in dimensions at least four

Determine a closed-form criterion for the existence of golden states under the Kraus-type superposition-free operations (FO) for overlap matrices of dimension d≥4 beyond the cases covered by Theorem GoldenFO.

Background

The paper establishes Gram majorization as an exact, target-by-target criterion for pure-state conversion under FO and derives complete existence results for several families of overlap matrices, including all three-reference-state configurations. However, for d≥4 the existence of a state from which every target can be reached remains unresolved in general. The authors explicitly identify the absence of a closed-form criterion beyond the cases treated in Theorem GoldenFO as an open problem.