Nonnegativity of genuine tri-entropy at n=1

Establish whether the genuine multi-entropy GM_1^{(3)}(A;B;C), defined via GM_n^{(3)} = S_n^{(3)} − (1/2)(S_n(A) + S_n(B) + S_n(C)) in the n→1 limit, is always nonnegative for arbitrary tripartite pure states in finite-dimensional Hilbert spaces.

Background

The genuine tri-entropy GM_1^{(3)} is proposed to capture intrinsic tripartite entanglement. For GHZ states, GM_1^{(3)} equals (1/2)·log k and is nonnegative, and all examples computed in the paper yield nonnegative values.

However, the general nonnegativity property has not been proven; clarifying it would solidify GM_1^{(3)} as a robust tripartite entanglement measure.