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.
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.
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 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.
The no-cloning theorem is a cornerstone result in quantum mechanics that forbids copying of general quantum information. More quantitatively, even given N copies of an unknown state, any channel producing an (N+1)-copy state incurs a nonzero trace-distance error on some inputs. Turning this around, one can ask about the sample complexity of approximate N→N+1 cloning: for a desired small error ε, what N suffices? For arbitrary pure states on a Hilbert space H, the answer is N≃dimH/ε - astronomically large for most H of interest. What if the input is promised to lie in a structured family? We examine a range of families fundamental to many-body physics and quantum information: n-mode fermionic Gaussian and Slater states, n-mode bosonic Gaussian states, and qudit phase states. For fermions, we construct optimal cloning channels, reducing the complexity from exponential to polynomial in n. For bosonic Gaussian states, we construct an explicit cloner that certifies polynomial complexity, without any constraint on the energy of the state; this channel, however, is not optimal in general. The fermionic and bosonic results follow from a single representation-theoretic framework, which generalizes previous work by Werner and by Chiribella and Yang; it also yields the cloners' unitary implementation. Furthermore, we explain why for many families the cloning complexity is linear in the dimension of the family manifold, via a controlled saddle-point evaluation of the frame potential at high N. The same calculation accounts for the analogous scaling of tomography complexity. The above framework does not cover phase states; for these we give an independent construction of a channel with optimal cloning fidelity. Our work complements recent results on cloning stabilizer states.
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.
Quantum state cloning is the task of approximately producing additional copies of an unknown quantum state from a finite number of input copies. The optimal cloning fidelity is known exactly for pure states, but no comparable characterization is known for general mixed states. We determine the optimal asymptotic copy complexity for d-dimensional states of rank at most r: producing M additional copies with worst-case fidelity at least 1−ε requires and is achievable with N=Θ(Mrd/ε) input copies. Remarkably, the lower bound already holds for a family of states with a fixed flat spectrum, while the matching upper bound is achieved by random purification followed by optimal pure-state cloning. For M=1, we further show that high-fidelity tomography can be coherently converted into cloning with comparable error, revealing an operational origin of the matching cloning and tomography complexities.
Werner's universal cloner turns T copies of an arbitrary pure state in C^d into T+k approximate copies with optimal global fidelity D_T/D_{T+k}, where D_m = dim Sym^m(C^d). When the input is drawn from a restricted ensemble, a better cloner may exist. We show that, for optimal average global fidelity, it exists unless the ensemble is a design of order T+k: for every probability measure on pure states and all T, k ≥ 1, the optimal value equals D_T/D_{T+k} if and only if the measure is a complex projective (T+k)-design. Sufficiency follows from known arguments; the new content is necessity, which we prove from semidefinite-programming complementary slackness together with a rigidity lemma for operators acting on symmetric tensors. As corollaries, Werner's cloner is optimal for the uniform ensemble of n-qubit stabilizer states exactly when T+k ≤ 3 (for every n), for symmetric informationally complete ensembles exactly when T+k ≤ 2, and for complete sets of mutually unbiased bases exactly when T+k ≤ 3 in d = 2 and T+k ≤ 2 in d ≥ 3. We give exact rational certificates for the first strict gaps, and we contrast the result with structured state families, where matching moments is neither necessary nor sufficient for optimality. Files: the paper (PDF) and its LaTeX source; paper2_certificates.zip, the exact-arithmetic certificate scripts with their outputs, README and SHA-256 checksums (MIT licence); SHA256SUMS.txt lists the SHA-256 hashes of all files. Companion paper: doi:10.5281/zenodo.23428376.