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.
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.
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.
The sample complexity of identifying an unknown pure stabilizer state has been a longstanding problem in quantum information. In this paper, we resolve the entire exact critical crossover for the uniform ensemble of pure n-qubit stabilizer states under arbitrary collective measurements and without side information. For k=n+c, with fixed integer c, the optimal success probability converges to an explicit positive series P∞(c). In particular, P∞(0)=0.1760837896…, while its two tails obey a quadratic-exponential law below threshold and 1−P∞(c)∼2−c above threshold. The pretty-good measurement is exactly optimal for every finite n and k. The proof diagonalizes the stabilizer Gram operator through Clifford tensor-power duality and a defect filtration, including the exceptional binary sectors. The resulting integer-scale law quantifies the n+O(1) copy threshold.
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 investigate the asymptotic relationship between quantum cloning and quantum estimation from the global point of view where all the copies produced by the cloner are considered jointly. For an N-to-M cloner, we consider the overall fidelity between the state of the M output systems and the state of M ideal copies, and we ask whether the optimal fidelity is attained by a measure and-prepare protocol in the limit M -> \infty. In order to gain intuition into the general problem, we analyze two concrete examples: i) cloning qubit states on the equator of the Bloch sphere and ii) cloning two-qubit maximally entangled states. In the first case, we show that the optimal measure-and-prepare fidelity converges to the fidelity of the optimal cloner in the limit M -> \infty. In the second case, we restrict our attention to economical covariant cloners, and again, we exhibit a measure-and-prepare protocol that achieves asymptotically the optimal fidelity. Quite counterintuitively, in both cases the optimal states that have to be prepared in order to maximize the overall fidelity are not product states corresponding to M identical copies, but instead suitable M-partite entangled states.
We determine asymptotic global fidelities of cloning generic mixed states with simple (non-degenerate) full-rank spectra. In fixed dimension d, let n inputs and mn outputs satisfy mn/n→γ>1. For a known simple full-rank spectrum p1>⋯>pd>0, we prove that the optimal root fidelity between the entire output and the product target state is $$
F_\rm{orb}(γ,p)
=\prod_{i<j}
\frac{\sqrtγ+\sqrt{q_{ij}(γ-1+q_{ij})}}
{γ+q_{ij}},
\qquad q_{ij}=\frac{p_j}{p_i}. $$ When the spectrum is unknown, one spectrum-independent cloner attains $$
F_\rm{univ}(γ,p)
=\left(\frac{2\sqrtγ}{1+γ}\right)^{(d-1)/2}
F_\rm{orb}(γ,p), $$ uniformly on compact subsets of the simple full-rank chamber and is minimax optimal. The two factors quantify spectral and eigenbasis uncertainty. Our channels combine classical processing of Schur--Weyl labels with a covariant Cartan-sector channel. Local asymptotic normality and a Gaussian fidelity bound against arbitrary channels yield the converses. For flat projector states ρ=P/r, with P a rank-r orthogonal projector, we separately prove the optimum γ−r(d−r)/2. Furthermore, we estimate the exact asymptotic fidelity of purify--clone--trace (PCT), and show that our cloners strictly outperform PCT at every fixed gain for simple full-rank spectra and rank-r projector states with 1<r<d.
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.