Coherent versus Measure-First Copying of Phase-State Codebooks: Exact Leak-Order Laws

Phase-state codebooks ψ_x = d-1/2 Σ_j ωx k_j |j⟩ are the states used in quantum fingerprinting and quantum hashing. We compare, for every copying task T→T+k, the optimal coherent copier (any quantum channel) and the optimal measure-first copier (entanglement-breaking: measure, then prepare) on a finite codebook with the same quantities for the continuous family of all phase states. Both finite-codebook optima are at least the continuous ones; if the keys have no additive collisions among multisets of size 2T+k, both are equal for every dimension; and for codebooks that are collision-free at size T, every collision at size 2T+k strictly increases the value of the natural measure-first strategy. For cyclic qubit codebooks of q states and every 1→m task (m ≥ 2) we prove an exact leak-order theorem: the measure-first optimum equals the continuous one if and only if q ≥ m+2, while the coherent optimum already equals it when q ≥ m+1, so q = m+1 separates the two attackers for every m, beginning with three qubit states at 1→2. For one input copy and two or three output copies we determine the measure-first optimum of the continuous family in every dimension d ≥ 3—for example (7d−8+√(25d²−64d+48))/(2d³) at 1→2—so that for cyclic codebooks in every dimension the coherent 1→2 optimum equals the continuous one if and only if the keys form a Sidon set, while the measure-first optimum does if and only if they form a B_3 set. We give an explicit polynomial-phase hash with a guaranteed (sufficient) copy budget and prove a separation for it. Whether measure-first attackers leak no later than coherent ones for every codebook remains open; the stronger statement that the measure-first gap always exceeds the coherent gap is false for general finite phase codebooks. Files: the paper (13 pages); hsep_qhn_certificates.zip — every certificate, independent checkers from three systems, expected logs and a SHA-256 manifest (SHA-256 2d9d96c45902b036bd1495569a0d972f5199567dc15fa22eb01611c633708413); paper4_source.zip — LaTeX source.