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 pose the question whether the asymptotic equivalence between quantum cloning and quantum state estimation, valid at the single-clone level, still holds when all clones are examined globally. We conjecture that the answer is affirmative and present a large amount of evidence supporting our conjecture, developing techniques to derive optimal asymptotic cloners and proving their equivalence with estimation in virtually all scenarios considered in the literature. Our analysis covers the case of arbitrary finite sets of states, arbitrary families of coherent states, arbitrary phase- and multiphase-covariant sets of states, and two-qubit maximally entangled states. In all these examples we observe that the optimal asymptotic cloners enjoy a universality property, consisting in the fact that scaling of their fidelity does not depend on the specific details of the input states, but only on the number of free parameters needed to specify them.
Perfect cloning of a known set of states with arbitrary prior probabilities is possible if we allow the cloner to sometimes fail completely. In the optimal case the probability of failure is at its minimum allowed by the laws of quantum mechanics. Here we show that it is possible to lower the failure rate below that of the perfect probabilistic cloner but the price to pay is that the clones are not perfect; the global fidelity is less than one. We determine the optimal fidelity of a cloner with a Fixed Failure Rate (FFR cloner) in the case of a pair of known states. Optimality is shown to be attainable by a measure-and-prepare protocol in the limit of infinitely many clones. The optimal protocol consists of discrimination with a fixed rate of inconclusive outcome followed by preparation of the appropriate clones. The convergence shows a symmetry-breaking second-order phase transition in the fidelity of the approximate infinite clones.
We investigate the probabilistic cloning and purification of quantum states. The performance of these probabilistic operations is quantified by the average fidelity between the ideal and actual output states. We provide a simple formula for the maximal achievable average fidelity and we explicitly show how to construct a probabilistic operation that achieves this fidelity. We illustrate our method on several examples such as the phase covariant cloning of qubits, cloning of coherent states, and purification of qubits transmitted via depolarizing channel and amplitude damping channel. Our examples reveal that the probabilistic cloner may yield higher fidelity than the best deterministic cloner even when the states that should be cloned are linearly dependent and are drawn from a continuous set.
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.
We consider the problem of approximate cloning of quantum states: given
n copies of an unknown state
ρ∈Cd×d, prepare an
(n+k)-copy state with high fidelity to
ρ⊗(n+k). Werner's pure state cloner is the optimal channel for the pure state case, and shows that
n=Θ(kd/ε) copies are necessary and sufficient to clone
k additional copies of an unknown pure state to fidelity
1−ε. The random purification channel gives a straightforward extension of Werner's cloner to mixed state inputs: given
n copies of a mixed state, randomly purify your input, apply Werner's channel in the larger Hilbert space, and then trace out the auxiliary registers. This gives a mixed state cloner using
n=O(krd/ε) copies to clone rank-
r states. Can one do any better? We show that the answer is no: one must use
n=Ω(krd/ε) copies. We prove our lower bound by studying the special case of projector cloning, in which the input state
ρ is promised to be of the form
P/r, where
P is a rank-
r orthogonal projector.
As a further application of our techniques, we consider the closely related problem of approximate transposition of quantum states, where one seeks to convert
ρ⊗n to a
k-copy state with high fidelity to
(ρT)⊗k. Here, we again show
n=Θ(krd/ε) copies are necessary and sufficient for this task.
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.