Phase-state codebooks ψ_x = d^{-1/2} Σ_j ω^{x 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.
Phase-state codebooks ψ_x = d^{-1/2} Σ_j ω^{x 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.
We determine exactly how well a measure-and-prepare ("measure-first") forger can copy qudit phase states. For one input copy and every number m of output copies, in every dimension d, we prove that the flat seed is the optimal measure-first strategy for the continuous family of phase states, the optimum being the top eigenvalue of an explicit matrix on the symmetric subspace (for example (7d − 8 + √(25d² − 64d + 48))/(2d³) at 1→2). The proof reduces the question to a negative-correlation inequality for closed lattice walks in symmetric time-dependent windows, which we prove by summation by parts; this combinatorial core is verified in the Lean 4 proof assistant. Consequently, for cyclic phase-state codebooks ψ_x = d^(−1/2) Σ_j ω^(x·k_j) |j⟩, the states used in quantum fingerprinting and quantum hashing, in every dimension and every 1→m task the measure-first optimum equals the continuous one if and only if the keys form a B_{m+1} set, whereas at 1→2 the coherent optimum does if and only if they form a Sidon set. Around this law we compare, for every copying task T→T+k, the optimal coherent copier (any quantum channel) and the optimal measure-first copier on a finite codebook with the continuous family: 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. Pre-registered experiments on IBM superconducting processors, in which the strategies are emulated by classically recombining two circuit stages, confirm the predicted qubit gaps at 1→2 and 1→3. 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.
Phase states |ψ_x⟩ = d^(−1/2) Σ_j ω^(x k_j) |j⟩, indexed by x ∈ Z_q and a key set K = {k_1,…,k_d} ⊂ Z_q, are the state family behind quantum fingerprinting and quantum hashing. We ask how well an adversary holding T copies of |ψ_x⟩ can produce T+k copies, and when a finite codebook is exactly as hard to clone as the continuous family of all phase states (the torus), whose optimal cloning fidelity was recently determined. The answer is governed by additive combinatorics of K. (i) The optimal 1→2 global cloning fidelity equals (2d−1)/d² if and only if K is a Sidon set. (ii) A B_{T+k} key set attains the torus optimum: analytically for d = 2 (all T) and for T = 1 (all d, k), and for T = 2, 3 (all d, k) by a proof whose finitely many computational steps are verified by standalone exact and interval-arithmetic checkers; we conjecture it for all T. (iii) For T = 1 we characterise exactly, for every k, the key sets that attain the torus optimum; the B_{1+k} condition is sufficient but not necessary. (iv) The optimal fidelity of fusing m phase states into the state of their sum equals E_m(K)/d^(m+1), where E_m is the higher additive energy; Sidon sets minimise it for every m. Small Fourier bias, the standard design criterion for quantum hashing, does not control cloning. For finite orbits of non-abelian groups we prove a window theorem giving a dual-feasibility criterion, an obstruction, and a representation-theoretic selection rule for when matching the (T+k)-th moment suffices, and we give certified counterexamples for SU(3) and SU(2) showing that moment matching alone does not. Changes in v2: new exact characterisation at T = 1 for every k (Section 5.4), including key sets that attain the torus value without being B_{T+k}; new m-fold fusion theorem with higher additive energy (Theorem 28); abstract and introduction updated accordingly. All v1 results are unchanged. Files: the paper (PDF) and its LaTeX source; ca_appendix.zip, the computer-assisted certificates with standalone exact/interval/ball-arithmetic checkers (unchanged from v1); coverage_check.py, the region-coverage audit (unchanged); qh-lab-v1.1.0.zip, an independent re-implementation with 49 tests (MIT licence). SHA256SUMS.txt lists the SHA-256 hashes of all files.
Copying an unknown quantum state into many approximate copies is known to become equivalent to estimating it and re-preparing: the best measure-and-prepare copier approaches the best quantum copier as the number of copies grows. We determine how fast. For equatorial qubit states, one input copy and m output copies, judged by the global fidelity, we prove F_EB/F_coh = 1 − 1/√(2m) + O(1/m). For phase states on d levels we prove, with explicit constants, the law 1 − F_EB/F_coh = (d − 1)/√(2m) + O(1/m), confirmed by exact computations up to m = 400; for qubits with T input copies the gap is √2·σ_T/√m, proved with explicit constants for T = 2, 3 (the upper bound conditionally on the optimality of the flat seed, R0(T); the lower bound unconditionally). The measure-and-prepare copier's deficit is the ground-state energy of a harmonic oscillator, one quantum per unknown phase. The exact quantum fidelity has plateaus at m = kd and kd + 1 (m = kd + 1 is the economical point). Numerical results suggest a period-d modulation of the ratio at order 1/m; a proof of its next-order coefficient remains open. This complements the known universality of the cloning fidelity itself, which depends only on the number of free parameters: within the phase-state family, the leading gap coefficient counts the independent phase directions. The classical values rest on the flat-seed theorem of the companion Paper 5, and the several-copy quantum values on the reduced coherent dual of the companion Paper 5. Core lemmas of the qubit proof and its explicit upper-bound constants are machine-checked in Lean 4; the remaining steps are proved on paper and independently refereed.
We determine the optimal average global fidelity of T→T+k approximate cloning for the uniform ensemble of n-qubit stabilizer states, for every number of qubits n, every k ≥ 1 and every 1 ≤ T ≤ 3. With D = 2^n, the optimum is P_stab(n;T,k) = C(D+T−1, T) · F_{T+k−κ(k)}, where F_s = ∏_{i=0}^{s−2}(2^i+1) / (D ∏_{i=0}^{s−2}(D+2^i)) is the stabilizer frame potential and κ(k) is the largest dimension of a doubly-even binary code of length k. The result rests on an exact formula for the largest eigenvalue of the mixed stabilizer moment operators: for all a+b ≥ 1 and all n it equals F_{a+b−w}, where w = min(a,b) + κ(|a−b|) is the Witt index of an associated Z4-valued quadratic form. We then ask what is lost by measuring first. For measure-and-prepare (entanglement-breaking) cloners we show that, after the pretty-good measurement, re-preparing a better output than copies of the guess helps exactly from four output copies on, because fourth powers erase the phases ±1, ±i of stabilizer amplitudes; that for 1→4 the optimum is 3(D+48+√(D²+16D+1664)) / ((D+1)(D+2)(D+4)(D+8)) for every n, so coherent cloning wins by less than a factor 5; and that for 2→4 there are two regimes: the pretty-good-measurement strategy is optimal for n ≤ 3, while for every n ≥ 4 Bell sampling is optimal, with fidelity 3/(D(D+2)) and a coherent advantage of exactly 5D/(D+4). The upper bounds are explicit dual certificates for the PPT relaxation, covering every n. Version 2 adds Part II (measure-and-prepare cloning), Part III (open problems), a certificates appendix, and citation fixes. Part I results and theorem numbering are unchanged from version 1. Files:• Weldon_2026_Exact_optimal_cloning_of_stabilizer_states_v2.pdf — the paper (15 pages).• paper3_v2_certificates.zip — every Part II certificate with standalone checkers, run scripts, expected logs and a SHA-256 manifest (SHA-256 236f7e4c5405e8e4030d476958e51aae5fb7b2fd2349c3ec9ddcb882cddf054f).• paper3_code.zip — Part I code and certificates, unchanged from version 1 (SHA-256 de2fc42b8bc4fa710b8506de2001d475a8d8b4a011ed0df72c693110a5d417e0).• paper3_v2_source.zip — LaTeX source.
Angle-encoded variational quantum circuits implement truncated Fourier series. For one-dimensional inputs, a single-qubit re-uploading (serial) circuit represents every function that a multi-qubit circuit with the same encoding budget E can; we find that this exchange of width for depth is not free in first-order training. Because its state lives on the Bloch sphere, all parameters of the serial circuit move the state, at each input, along one complex direction, which we call \emph{phase-locking}. We prove that this costs no Fourier direction but leaves no direction to spare: however many parameters it has, the serial circuit has at most 2E+2 tangent directions, one more than its 2E+1 Fourier coefficients need, while a two-qubit control with the same encoding budget and model class generically has at least 6E+6. The cost in training is measured against this control, as the number of runs out of 100 that reach the optimum within a fixed step budget. On an exact loss with flat-spectrum targets the gap grows with E; at E=32 the serial circuit converges in 3, 53 and 83 runs under gradient descent, Adam and Levenberg--Marquardt, the control in 99, 100 and 100. On three real-world series, at matched parameter counts, the serial circuit reaches the least-squares optimum in at most 9 runs under Adam and in 32 to 93 under Levenberg--Marquardt, the control in at least 98 and in all 100; its held-out error is up to about 1.5 times that of the control under Adam, but no larger in the typical run under Levenberg--Marquardt.
Abstract Copying is the most basic operation on information, and the no-cloning theorem is the sharpest statement of how quantum information differs from classical. The best approximate copier of pure states has been known for decades, and the asymptotic theory of cloning mixed states at fixed gain γ=m/n was recently completed by Schur--Weyl methods, which give closed-form limiting fidelities and an asymptotically optimal ``Schur--Cartan'' cloner. Whether those constructions are also good at the finite copy numbers available in experiments, and what the finite-copy optimum is, was unknown. We show that the finite-copy optimum is a concave program over Young-label transition rules and extremal covariant maps between irreducible sectors, solve it exactly for qubits up to n=40 and for qutrits and ququarts at small n, and show that the asymptotically optimal cloners lose up to 0.11 in root fidelity at n≤20, while purify--clone--trace fails at the degenerate spectrum. The loss is entirely in the label rule: the Cartan sector channel is finite-copy optimal for qubits, and in higher dimension it coincides with the lowest-Casimir (minimal Parthasarathy--Ranga Rao--Varadarajan) component, which is within 10−3 of optimal. A closed-form cloner, a Knothe--Rosenblatt coupling of the input and target label laws followed by lowest-Casimir channels, is asymptotically optimal, improves the Schur--Cartan cloner by up to 0.115, and comes with Karush--Kuhn--Tucker and Frank--Wolfe certificates bracketing the true optimum to 10−3 without semidefinite programming. The analysis isolates a single representation-theoretic question, the pairwise optimality of the PRV channel at intermediate spectral mixedness, settled here in the cold and hot limits and for single-copy inputs, which governs how much coherent cloning advantage survives at the copy numbers of near-term quantum networks.