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 $m_n$ outputs satisfy $m_n/n\toγ>1$. For a known simple full-rank spectrum $p_1>\cdots>p_d>0$, 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 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 $$
F_\rm{orb}(γ,p), $$ 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.
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.
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.
In this work, we study asymmetric (1→2) quantum cloning for pure states belonging to orbits generated by the orthogonal group. For every dimension (d≥3), we determine the full region of achievable pairs of average single-copy fidelities. Using orthogonal covariance and the Brauer algebra, we reduce the optimization over all CPTP maps to a finite-dimensional spectral problem and construct explicit optimal cloning channels. We then apply the general results to four-mode fermionic states in the even-parity sector. Using triality, we identify the fixed-concurrence fermionic families with the orthogonal orbits considered in the first part of the paper. In particular, for pure Gaussian states we show that the symmetric locally optimal cloner is unique. We also compare local and global cloning and find that the channel optimal for the joint two-copy fidelity is different from the one optimal for the single-copy fidelities. Finally, we show that the locally optimal cloner cannot be implemented using only operations which preserve fermionic Gaussian states.
We study the number of copies needed to estimate the root Uhlmann fidelity between an unknown quantum state and a classically known reference, under collective measurements. If the unknown state has rank at most s, we give an estimator using O(s2/ε2) copies, uniformly in the ambient dimension and reference rank. The estimator applies a random-purification channel and a covariant pure-state measurement, then rescales the observed amplitude before evaluating a weighted nuclear norm. Combined with the lower bound established independently in our first version and in concurrent work of Wang, this determines the worst-case complexity under rank bounds r,s as min{r,s}2/ε2 up to logarithmic factors in the smaller rank.
The same estimator gives the upper bound O(r(trσ)2/ε2) for a rank-r reference σ. We combine spectral truncation with lower bounds obtained by embedding hard instances for the maximally mixed reference into spectral subspaces. For spectra λi∝i−α with fixed 1<α≤2, as ε↓0 with r≥Cαε−2/(α−1), the bounds determine the complexity as Θα(ε−4/(α−1)). In particular, inverse-square spectra have accuracy exponent four. The lower-bound construction uses exact moment matching and an explicitly computable Schur measure.
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.