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Finite-copy optimal cloning of mixed qudit states:\\ exact optima, a closed-form cloner, and the structure of covariant sector channels

Hikaru Wakaura

Published Oct 26, 2026• 0 citations
Publisher

Abstract

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 $\gamma=m/n$ 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 $n\le 20$, 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.

DOI
10.21203/rs.3.rs-11480035/v1

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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\gamma=m/nγ=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=40n=40n=40 and for qutrits and ququarts at small nnn, and show that the asymptotically optimal cloners lose up to 0.110.110.11 in root fidelity at n≤20n\le 20n≤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 10−310^{-3}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 0.1150.1150.115, and comes with Karush--Kuhn--Tucker and Frank--Wolfe certificates bracketing the true optimum to 10−310^{-3}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.
We determine asymptotic global fidelities of cloning generic mixed states with simple (non-degenerate) full-rank spectra. In fixed dimension ddd, let nnn inputs and mnm_nmn​ outputs satisfy mn/n→γ>1m_n/n\toγ>1mn​/n→γ>1. For a known simple full-rank spectrum p1>⋯>pd>0p_1>\cdots>p_d>0p1​>⋯>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/rρ=P/rρ=P/r, with PPP a rank-rrr orthogonal projector, we separately prove the optimum γ−r(d−r)/2γ^{-r(d-r)/2}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-rrr projector states with 1<r<d1<r<d1<r<d.
We consider the problem of approximate cloning of quantum states: given nnn copies of an unknown state ρ∈Cd×dρ\in \mathbb{C}^{d \times d}ρ∈Cd×d, prepare an (n+k)(n+k)(n+k)-copy state with high fidelity to ρ⊗(n+k)ρ^{\otimes (n+k)}ρ⊗(n+k). Werner's pure state cloner is the optimal channel for the pure state case, and shows that n=Θ(kd/ε)n = Θ(kd/\varepsilon)n=Θ(kd/ε) copies are necessary and sufficient to clone kkk additional copies of an unknown pure state to fidelity 1−ε1-\varepsilon1−ε. The random purification channel gives a straightforward extension of Werner's cloner to mixed state inputs: given nnn 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)n = O(krd/\varepsilon)n=O(krd/ε) copies to clone rank-rrr states. Can one do any better? We show that the answer is no: one must use n=Ω(krd/ε)n = Ω(krd/\varepsilon)n=Ω(krd/ε) copies. We prove our lower bound by studying the special case of projector cloning, in which the input state ρρρ is promised to be of the form P/rP/rP/r, where PPP is a rank-rrr 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⊗nρ^{\otimes n}ρ⊗n to a kkk-copy state with high fidelity to (ρT)⊗k(ρ^T)^{\otimes k}(ρT)⊗k. Here, we again show n=Θ(krd/ε)n = Θ(krd/\varepsilon)n=Θ(krd/ε) copies are necessary and sufficient for this task.
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 ddd-dimensional states of rank at most rrr: producing MMM additional copies with worst-case fidelity at least 1−ε1-\varepsilon1−ε requires and is achievable with N=Θ(Mrd/ε)N=Θ(Mrd/\varepsilon)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=1M=1M=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.
The no-cloning theorem is a cornerstone result in quantum mechanics that forbids copying of general quantum information. More quantitatively, even given NNN copies of an unknown state, any channel producing an (N+1)(N+1)(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+1N\rightarrow N+1N→N+1 cloning: for a desired small error εεε, what NNN suffices? For arbitrary pure states on a Hilbert space HHH, the answer is N≃dim⁡H/εN\simeq \dim H/εN≃dimH/ε - astronomically large for most HHH 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: nnn-mode fermionic Gaussian and Slater states, nnn-mode bosonic Gaussian states, and qudit phase states. For fermions, we construct optimal cloning channels, reducing the complexity from exponential to polynomial in nnn. 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 NNN. 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.
Abstract Copying quantum information is forbidden by the no-cloning theorem, yet approximate copies with the highest fidelity allowed by quantum mechanics are a basic resource for quantum networks and for assessing the security of quantum key distribution. Such cloners are usually gate sequences, but a spin network with fixed couplings can perform the transformation autonomously: the unknown qubit and blank qubits sit on the vertices of a graph and the free evolution does the rest. Until now, the only spin graph known to reach the optimal phase-covariant fidelity for every number of clones MMM was the star with a tuned anisotropy. Which other graphs can clone optimally, and why, has remained open. Here we show that the complete bipartite graph K2,M−1K_{2,M-1}K2,M−1​---the unknown qubit and one target on one side, the remaining targets on the other---reaches the optimal 1→M1\to M1→M phase-covariant fidelity for every M≥3M\ge 3M≥3, with all couplings given in closed form. For odd MMM a staggered Ising anisotropy suffices, with the parameters fixed by two integers; for even MMM an additional Ising coupling among the middle targets is required, with its value an explicit algebraic number. Both families reduce, in the permutation-symmetric sectors, to two four-level problems: a perfect-state-transfer ladder on a four-cycle and a freeze block whose characteristic polynomial we obtain analytically. We verify the formulas with no free parameter up to M=10M=10M=10 and classify exactness for all four-vertex graphs with a least-squares test of the exact revival and transfer equations. The results connect the fractional-revival engineering of spin chains to optimal cloning and identify the hidden involution behind both. Beyond cloning, optimal quantum machines can thus be read off from the spectrum of small symmetric blocks, suggesting autonomous implementations of other covariant transformations on spin and Rydberg arrays.
In this work, we study asymmetric (1→2)(1\to2)(1→2) quantum cloning for pure states belonging to orbits generated by the orthogonal group. For every dimension (d≥3)(d\geq3)(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.