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arXiv:2603.23628

Optimal pure state cloning and transposition are complementary channels

Vanessa Brzić, Dmitry Grinko, Michał Studziński, Marco Túlio Quintino

quant-phSubmitted Mar 24, 2026• Announced Mar 26, 2026• 0 citations• Revised Mar 26, 2026 (v2)
v1 · Mar 24, 2026v2 · Mar 26, 2026v3 · Sep 26, 2026
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Abstract

State cloning and state transposition are fundamental transformations which, despite being desirable, cannot be perfectly realised due to two conceptually distinct constraints of quantum theory: cloning is forbidden by linearity, while transposition is ruled out by complete positivity. In this work, we show that, despite these different constraints, 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. We first determine the optimal fidelity for transforming $N$ qudits into $K$ copies of their transposition and show that, for pure input states, it is achieved by an estimation strategy, which is the unique optimal strategy under the worst-case fidelity figure of merit. We further prove that the corresponding $N \to K$ transposition map is the complementary channel of the optimal universal symmetric $N \to N + K$ quantum cloning machine on pure states. We then present an explicit quantum circuit that realises $N \to K$ transposition and $N \to N + K$ cloning in parallel and analyse its gate efficiency. Finally, we investigate mixed-state $N \to 1$ qudit transposition and determine its maximal performance in terms of white-noise visibility, yielding the structural physical approximation of transposition in the multicopy regime.

Comments
24 pages total (5 pages main text, 19 pages appendix), 1 figure

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State cloning and state transposition are fundamental transformations which, despite being desirable, cannot be perfectly realised due to two conceptually distinct constraints of quantum theory: cloning is forbidden by linearity, while transposition is ruled out by complete positivity. In this work, we show that, despite these different constraints, 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. We first determine the optimal fidelity for transforming NNN qudits into KKK copies of their transposition and show that, for pure input states, it is achieved by an estimation strategy, which is the unique optimal strategy under the worst-case fidelity figure of merit. We further prove that the corresponding N→KN \to KN→K transposition map is the complementary channel of the optimal universal symmetric N→N+KN \to N + KN→N+K quantum cloning machine on pure states. We then present an explicit quantum circuit that realises N→KN \to KN→K transposition and N→N+KN \to N + KN→N+K cloning in parallel and analyse its gate efficiency. Finally, we investigate mixed-state N→1N \to 1N→1 qudit transposition and determine its maximal performance in terms of white-noise visibility, yielding the structural physical approximation of transposition in the multicopy regime.
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.
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.
We prove quantitative limitations on any approximate simultaneous cloning or broadcasting of mixed states. The results are based on information-theoretic (entropic) considerations and generalize the well-known no-cloning and no-broadcasting theorems. We also observe and exploit the fact that the universal cloning machine on the symmetric subspace of nnn qudits and symmetrized partial trace channels are dual to each other. This duality manifests itself both in the algebraic sense of adjointness of quantum channels and in the operational sense that a universal cloning machine can be used as an approximate recovery channel for a symmetrized partial trace channel and vice versa. The duality extends to give control of the performance of generalized universal quantum cloning machines (UQCMs) on subspaces more general than the symmetric subspace. This gives a way to quantify the usefulness of a priori information in the context of cloning. For example, we can control the performance of an antisymmetric analog of the UQCM in recovering from the loss of $n\ensuremath{-}k$ fermionic particles.
While the no-cloning theorem, which forbids the perfect copying of quantum states, is well-known as one of the defining features of quantum mechanics, the question of how well the theory allows a state to be cloned is yet to be completely solved. In this paper, rigorous solutions to the problem of M→NM\rightarrow NM→N asymmetric cloning of qudits are obtained in a number of interesting cases. The central result is the solution to the 1→N1 \rightarrow N1→N universal asymmetric qudit cloning problem for which the exact trade-off in the fidelities of the clones for every NNN and ddd is derived. Analogous results are proven for qubits when M=N−1M=N-1M=N−1. We also consider state-dependent 1→N1 \rightarrow N1→N qubit cloning, providing a general parametrization in terms of a Heisenberg star Hamiltonian. In all instances, we determine the feasibility of implementing the cloning economically, i.e., without an ancilla, and determine the dimension of the ancilla when an economic implementation is not possible.
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.
While the no-cloning theorem, which forbids the perfect copying of quantum states, is well-known as one of the defining features of quantum mechanics, the question of how well the theory allows a state to be cloned is yet to be completely solved. In this paper, rigorous solutions to the problem of M to N asymmetric cloning of qudits are obtained in a number of interesting cases. The central result is the solution to the 1 to N universal asymmetric qudit cloning problem for which the exact trade-off in the fidelities of the clones for every N and d is derived. Analogous results are proven for qubits when M=N-1. We also consider state-dependent 1 to N qubit cloning, providing a general parametrization in terms of a Heisenberg star Hamiltonian. In all instances, we determine the feasibility of implementing the cloning economically, i.e., without an ancilla, and determine the dimension of the ancilla when an economic implementation is not possible.