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arXiv:2608.27298

Optimal cloning of mixed states

Marco Fanizza, Dmitry Grinko, Thilo Scharnhorst, Jack Spilecki

quant-phSubmitted Aug 27, 2026• Announced Aug 28, 2026• 0 citations
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Abstract

We consider the problem of approximate cloning of quantum states: given $n$ copies of an unknown state $ρ\in \mathbb{C}^{d \times d}$, prepare an $(n+k)$-copy state with high fidelity to $ρ^{\otimes (n+k)}$. Werner's pure state cloner is the optimal channel for the pure state case, and shows that $n = Θ(kd/\varepsilon)$ copies are necessary and sufficient to clone $k$ additional copies of an unknown pure state to fidelity $1-\varepsilon$. 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 $n = O(krd/\varepsilon)$ copies to clone rank-$r$ states. Can one do any better? We show that the answer is no: one must use $n = Ω(krd/\varepsilon)$ 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 $ρ^{\otimes n}$ to a $k$-copy state with high fidelity to $(ρ^T)^{\otimes k}$. Here, we again show $n = Θ(krd/\varepsilon)$ copies are necessary and sufficient for this task.

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Optimal copy complexity of quantum state cloning

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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 $1-\varepsilon$ requires and is achievable with $N=Θ(Mrd/\varepsilon)$ 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.

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Asymptotically Optimal Mixed-State Cloning

Jiani Fei, Jinzhao Wang

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 $1<r<d$.

quant-pharXiv:2609.35986•Sep 30, 2026•0 citations

Approximate cloning of structured pure states

Yaroslav Herasymenko, Junqiao Lin, Philip Verduyn Lunel +1 more

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 $N\rightarrow N+1$ cloning: for a desired small error $ε$, what $N$ suffices? For arbitrary pure states on a Hilbert space $H$, the answer is $N\simeq \dim H/ε$ - 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.

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Optimal physical approximations of pure-state cloning and transposition are complementary channels

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State cloning and state transposition are fundamental transformations that, despite being desirable, cannot be perfectly realised as a consequence of two distinct requirements of quantum theory: cloning is forbidden by linearity, while transposition is ruled out by complete positivity. In this work, we show that, in spite of these disparate conceptual restrictions, the best physically allowed realisation of both transformations arises from a single physical process described by an isometry, which simultaneously implements their best possible approximations. This complementary relation allows us to determine the exact quantum and private classical capacities of the universal symmetric $N\to N+K$ cloning channel in the fully general multicopy qudit setting, thereby closing the remaining gap in the quantum capacity problem for universal cloning. We provide an explicit circuit that simultaneously implements optimal pure-state $N\to K$ transposition and optimal $N\to N+K$ symmetric cloning, both attaining the same fidelity. Additionally, we prove that the optimal $N\to K$ pure-state transposition channel is unique and realised by an estimation strategy. We also obtain the optimal mixed-state $N\to 1$ transposition, and discuss its differences from its pure-state analogue.

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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

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.

•Oct 26, 2026•0 citations

Random purification channel for passive Gaussian bosons

Francesco Anna Mele, Filippo Girardi, Senrui Chen +2 more

The random purification channel, which, given $n$ copies of an unknown mixed state $ρ$, prepares $n$ copies of an associated random purification, has proved to be an extremely valuable tool in quantum information theory. In this work, we construct a Gaussian version of this channel that, given $n$ copies of a bosonic passive Gaussian state, prepares $n$ copies of one of its randomly chosen Gaussian purifications. The construction has the additional advantage that each purification has a mean photon number which is exactly twice that of the initial state. Our construction relies on the characterisation of the commutant of passive Gaussian unitaries via the representation theory of dual reductive pairs of unitary groups.

quant-pharXiv:2512.16878•Dec 19, 2025•0 citations
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.
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.
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.
State cloning and state transposition are fundamental transformations that, despite being desirable, cannot be perfectly realised as a consequence of two distinct requirements of quantum theory: cloning is forbidden by linearity, while transposition is ruled out by complete positivity. In this work, we show that, in spite of these disparate conceptual restrictions, the best physically allowed realisation of both transformations arises from a single physical process described by an isometry, which simultaneously implements their best possible approximations. This complementary relation allows us to determine the exact quantum and private classical capacities of the universal symmetric N→N+KN\to N+KN→N+K cloning channel in the fully general multicopy qudit setting, thereby closing the remaining gap in the quantum capacity problem for universal cloning. We provide an explicit circuit that simultaneously implements optimal pure-state N→KN\to KN→K transposition and optimal N→N+KN\to N+KN→N+K symmetric cloning, both attaining the same fidelity. Additionally, we prove that the optimal N→KN\to KN→K pure-state transposition channel is unique and realised by an estimation strategy. We also obtain the optimal mixed-state N→1N\to 1N→1 transposition, and discuss its differences from its pure-state analogue.
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.
The random purification channel, which, given nnn copies of an unknown mixed state ρρρ, prepares nnn copies of an associated random purification, has proved to be an extremely valuable tool in quantum information theory. In this work, we construct a Gaussian version of this channel that, given nnn copies of a bosonic passive Gaussian state, prepares nnn copies of one of its randomly chosen Gaussian purifications. The construction has the additional advantage that each purification has a mean photon number which is exactly twice that of the initial state. Our construction relies on the characterisation of the commutant of passive Gaussian unitaries via the representation theory of dual reductive pairs of unitary groups.