Prove permutational symmetry of QFI-optimal initial states

Prove that initial states maximizing the quantum Fisher information for local spontaneous-emission noise and phase estimation can be chosen within the permutationally symmetric subspace.

Background

The numerical optimization restricts the initial atomic-ensemble states to the completely symmetric subspace, which makes the spontaneous-emission dynamics computationally tractable. This restriction is physically motivated because the noise channel acts identically on every atom, and numerical results indicate that symmetric states perform substantially better than antisymmetric or mixed-symmetry states.

Nevertheless, the paper explicitly notes that no rigorous result establishes that quantum-Fisher-information-maximizing states for local qubit channels must possess the same permutational symmetry as the channel. Establishing such a theorem would justify the global-optimality claims beyond the numerically explored symmetric sector.