How much of F_Q can position read?

A phase θ is written by transport, e−iθT, and read with position P. The quantum Fisher information is FQ = 4 Var T; reading with P gives FP = |⟨[T, P]⟩|² / Var P. Lean: FP ≤ FQ, always. Turn the phase of the state and watch the two fluctuation vectors open and close.

The state is ψj ∝ sin((j+1)π/(d+1)) e−ijφ; at φ = π/2 it is the maximal current state of NRS, where the efficiency is exactly 1/CNava(d)² = cos² θNRS(d): 1 at d = 2, 3 and strictly below from d = 4 on (D41 in nava-robertson-schrodinger). The bound itself is CramerRao.fisher_le_qfi, from Robertson (CramerRao.robertson) — source. Grey dots: random states of the same d.