Why precision crawls: the 1/√N wall

Each flip has variance p(1−p), largest at p=½ and zero at p=0 or 1. Now read it as a qubit measured in the {|0⟩,|1⟩} basis. If the qubit is already an eigenstate of what you measure (|0⟩ or |1⟩), the projection noise √(p(1−p)) vanishes — the observable already has a definite value in this state, so repeating just remeasures the same thing (a two-headed coin) and adds no information. A superposition collapses to a random outcome with probability p = |⟨1|ψ⟩|² — a coin whose bias is the state's overlap with an eigenstate. The equal superposition |+⟩ is the fair coin: maximally random, maximal projection noise. The standard quantum limit, 1/√N, is exactly this projection noise piled up over N independent measurements — and it bites hardest where you actually sense, in superposition.

Bernoulli per trial: Var = p(1−p) (max ¼ at p=½). The estimate p̂ = count/N has standard error σ = √(p(1−p)/N); the shape is the prefactor √(p(1−p)) (plot ②), the scaling is N^−1/2 (plot ③). For N qubits the count is Binomial(N,p) with the same Var = Np(1−p) — "quantum projection noise" (Itano et al., Phys. Rev. A 47, 3554 (1993)). A Ramsey phase is read at the equal-superposition point, where sensitivity and projection noise are both maximal, giving Δφ_SQL ∝ N^−1/2. Boundary: the 1/√N scaling is classical counting — not uniquely quantum; only the "fairness" (the overlap) is. Beating it needs correlations — squeezing, entanglement (anchors 4 & 5).