# Quantum cloning
**Quantum cloning** is the approximate copying of unknown quantum states by a physical device called a quantum cloning machine. Perfect copying of an arbitrary unknown quantum state is forbidden by the no-cloning theorem, so cloning machines are transformations that achieve the best approximate copying compatible with that theorem.[1](https://arxiv.org/abs/quant-ph/0210061) The quality of a copy is measured by its fidelity, the overlap between the copy and the original state, which equals 1 for a perfect copy and 0 for orthogonal states.[2](https://qiskit.qotlabs.org/learning/courses/general-formulation-of-quantum-information/purifications-and-fidelity/fidelity)
| Key fact | Detail |
|---|---|
| No-cloning theorem | The linearity of quantum mechanics forbids perfect replication of an arbitrary unknown quantum state (Wootters and Zurek, 1982)[3](https://doi.org/10.1038/299802A0) |
| Universal 1→2 qubit cloner | Bužek and Hillery (1996) introduced a universal quantum copying machine whose output quality does not depend on the input[4](https://arxiv.org/abs/quant-ph/9607018); its optimal fidelity is 5/6[5](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.57.2368) |
| N→M universal fidelity | For N identical input qubits cloned into M copies, the optimal fidelity is F = (NM+N+M)/(M(N+2))[6](https://harvest.aps.org/v2/journals/articles/10.1103/PhysRevLett.81.2598/fulltext) |
| Probabilistic cloning | States secretly chosen from a set can be cloned with unit fidelity if and only if the states are linearly independent (Duan and Guo, 1998)[7](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.80.4999) |
| Continuous-variable cloner | The Gaussian cloner copies any two canonically conjugate variables equally well and duplicates all coherent states with fidelity F = 2/3[1](https://arxiv.org/abs/quant-ph/0210061) |
| Coherent-state N→M fidelity | The optimal N-to-M cloning fidelity for coherent states is MN/(MN+M−N)[8](https://arxiv.org/abs/quant-ph/0005044) |
| Cryptographic role | Using an optimal cloner yields a tight bound on the best individual eavesdropping strategy in a quantum cryptosystem[1](https://arxiv.org/abs/quant-ph/0210061) |
## The no-cloning theorem
The starting point is a negative result. Wootters and Zurek showed in 1982 that the linearity of quantum mechanics forbids such replication, and that this conclusion holds for all quantum systems.[3](https://doi.org/10.1038/299802A0) The standard proof checks that a cloning transformation cannot preserve inner products: if a device cloned all states, the inner product between two input states would have to equal the inner product of their clones, which fails for nonorthogonal states.[9](https://math.mit.edu/~shor/18.435/oldlectures/lecture5.html) The result extends to mixed states: if a particle in an arbitrary mixed state enters a device and two particles emerge, the two reduced density matrices cannot both be identical to the input density matrix.[4](https://arxiv.org/abs/quant-ph/9607018)
The theorem leaves open a quantitative question: when ideal copies are unavailable, how close to the original can a copy be?[4](https://arxiv.org/abs/quant-ph/9607018) Answering this question defines the field of quantum cloning machines.
## Universal qubit cloners
**Bužek and Hillery.** In 1996, Vladimir Bužek and Mark Hillery analyzed to what extent arbitrary states of a two-level system can be copied and showed that a universal quantum copying machine exists, one that approximately copies quantum states so that the quality of its output does not depend on the input.[4](https://arxiv.org/abs/quant-ph/9607018) They also examined a machine combining a unitary transformation with a selective measurement that produces good copies of states in a neighborhood of a particular state.[4](https://arxiv.org/abs/quant-ph/9607018)
**Optimal fidelities.** Bruß, DiVincenzo, Ekert, Fuchs, Macchiavello, and Smolin established in 1998 that the maximal fidelity for universal 1→2 qubit cloners is 5/6, achievable either by a special unitary evolution or by a teleportation scheme.[5](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.57.2368) A single-copy fidelity of 5/6 indicates that, upon measurement in the basis of the original qubit, the clone will collapse to that qubit's state with probability 5/6.[10](https://journals.aps.org/prresearch/pdf/10.1103/PhysRevResearch.5.013139) For the general case, Gisin and Massar presented machines that transform N identical qubits into M identical copies and proved the copy fidelity optimal; as M tends to infinity, the fidelity of each clone tends toward the best fidelity obtainable by a measurement on the input qubits, so cloning machines are universal devices for translating quantum information into classical information.[11](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.79.2153) The tight N→M bound is F = (NM+N+M)/(M(N+2)), and M clones can be produced either directly or through intermediate steps, both giving the same overall shrinking factor.[6](https://harvest.aps.org/v2/journals/articles/10.1103/PhysRevLett.81.2598/fulltext) Werner constructed the unique optimal device for turning a finite number of d-level systems in the same unknown pure state into M systems approximating the M-fold tensor product of that state.[12](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.58.1827)
**State-dependent and asymmetric cloners.** A machine tailored to a restricted set of inputs can beat the universal bound: the phase-covariant cloning machine, which copies equatorial qubit states, reaches a fidelity of 0.854, higher than the universal 5/6, and provides the optimal eavesdropping method for a large class of attacks on quantum cryptographic protocols.[13](https://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.99.012324/fulltext) In asymmetric cloning, the two output copies have different qualities, and an exact trade-off between the clone fidelities has been derived for 1→N universal qudit cloning for every N and dimension d.[14](https://arxiv.org/abs/1208.5574) This trade-off shows that two close-to-perfect copies cannot be achieved simultaneously, a constraint imposed by quantum mechanics.[15](https://pmc.ncbi.nlm.nih.gov/articles/PMC4258659/)
## Probabilistic cloning
Deterministic cloners always produce approximate copies. A different route is to accept a chance of failure. Using a general operation that combines a unitary transformation with a reduction, Duan and Guo built a quantum cloning device that works probabilistically: after the measurement outcomes are postselected, it produces exact copies of the states fed into it.[7](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.80.4999) They proved that states secretly chosen from a set can be probabilistically cloned if and only if the states are linearly independent, and derived the best possible cloning efficiencies.[7](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.80.4999) If the states are linearly dependent, the linearity of quantum mechanics forbids such replication, just as in the proof of the no-cloning theorem.[16](https://arxiv.org/abs/quant-ph/9705018)
## Continuous-variable cloning
Cloning also applies to systems with a continuous spectrum, such as the quadrature components x and p of a light mode, which behave like position and momentum with commutator [x, p] = i. It is impossible to copy perfectly the eigenstates of two conjugate variables: a cloner that copies localized position states exactly fails for momentum states, and a plane-wave cloner fails for position states.[1](https://arxiv.org/abs/quant-ph/0210061)
[Nicolas J. Cerf](https://www.edgepedia.org/nicolas-j-cerf) introduced a Gaussian cloning transformation that copies equally well any two canonically conjugate continuous variables and duplicates all coherent states with the same fidelity, F = 2/3.[1](https://arxiv.org/abs/quant-ph/0210061) This fidelity is independent of the coherent-state amplitude α, so the cloner copies all coherent states equally.[17](https://citeseerx.ist.psu.edu/document?doi=8da2dc625da2d48d7cce3b350e849980e25d1212&repid=rep1&type=pdf) The optimal N-to-M cloning fidelity for coherent states is MN/(MN+M−N).[8](https://arxiv.org/abs/quant-ph/0005044)
**Gaussian versus non-Gaussian.** The Gaussian cloner is not optimal under every figure of merit. With joint fidelity as the measure, a Gaussian cloner maximizes the copying quality, but with single-clone fidelity the optimum is reached by a non-Gaussian cloner: the optimal single-clone fidelity for a symmetric 1-to-2 cloner is 0.6826, above the Gaussian value of 2/3.[18](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.95.070501)
## Experimental realizations
Several physical platforms have implemented cloning machines. Huang, Li, and colleagues at the [University of Science and Technology of China](https://www.edgepedia.org/university-of-science-and-technology-of-china) realized the 1→2 qubit Bužek-Hillery machine with linear optical devices, representing several qubits by a single photon, and measured fidelities of 5/6 for arbitrary input pure states.[19](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.64.012315)
A more direct route uses stimulated emission. Lamas-Linares, Simon, Howell, and Bouwmeester sent a single input photon into a parametric down-conversion source, where it stimulated the emission of additional photons; the unavoidable presence of spontaneous emission sets the fundamental quantum limit on clone quality, and the experiment produced clones with near-optimal fidelity, demonstrating universality by achieving the same fidelity for arbitrary input states.[20](https://arxiv.org/abs/quant-ph/0205149) In that scheme the fidelity works out as F = (2/3) × 1 + (1/3) × (1/2) = 5/6, the optimal achievable fidelity for universal cloning of a single photon.[20](https://arxiv.org/abs/quant-ph/0205149) The link between optimal cloning and light amplification via stimulated emission is a recognized part of the theory, alongside experimental demonstrations of optimal cloning.[21](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.77.1225)
## Role in quantum cryptography
The main application is security analysis. An optimal cloner typically allows one to derive a tight limit on the most effective individual attack an eavesdropper can mount against a quantum cryptosystem, and this link inspired research into continuous-variable cloning for quantum key distribution protocols that use continuous Gaussian key carriers.[1](https://arxiv.org/abs/quant-ph/0210061) The connection is precise for some protocols: for quantum key distribution protocols with 2 or d+1 [mutually unbiased bases](https://www.edgepedia.org/mutually-unbiased-bases), the optimal collective eavesdropping attack is an optimal cloner, while for protocols with d mutually unbiased bases it is not.[22](https://doi.org/10.1103/PhysRevA.85.052310) In the error-free case, the optimal individual attack on the [BB84 protocol](https://www.edgepedia.org/bb84-protocol) is given by phase-covariant cloning machines.[23](https://www.nature.com/articles/s41534-025-01088-9)
## References
1. [Nicolas J. Cerf (2002). Quantum cloning with continuous variables. arXiv.](https://arxiv.org/abs/quant-ph/0210061)
2. [Fidelity, IBM Quantum Documentation](https://qiskit.qotlabs.org/learning/courses/general-formulation-of-quantum-information/purifications-and-fidelity/fidelity)
3. [Wootters, W., Zurek, W. (1982). A single quantum cannot be cloned. Nature 299, 802–803.](https://doi.org/10.1038/299802A0)
4. [Bužek, V., Hillery, M. (1996). Quantum copying: Beyond the no-cloning theorem. arXiv.](https://arxiv.org/abs/quant-ph/9607018)
5. [Bruß, D., DiVincenzo, D. P., Ekert, A., Fuchs, C. A., Macchiavello, C., Smolin, J. A. (1998). Optimal universal and state-dependent quantum cloning. Phys. Rev. A 57, 2368.](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.57.2368)
6. [Bruß, D., Ekert, A., Macchiavello, C. (1998). Optimal universal quantum cloning and state estimation. Phys. Rev. Lett. 81, 2598 (full text).](https://harvest.aps.org/v2/journals/articles/10.1103/PhysRevLett.81.2598/fulltext)
7. [Duan, L.-M., Guo, G.-C. (1998). Probabilistic cloning and identification of linearly independent quantum states. Phys. Rev. Lett. 80, 4999.](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.80.4999)
8. [Optimal N-to-M cloning of quantum coherent states. arXiv.](https://arxiv.org/abs/quant-ph/0005044)
9. [18.435 Quantum Computation, Lecture 5, MIT (Peter Shor)](https://math.mit.edu/~shor/18.435/oldlectures/lecture5.html)
10. [Quantum-optical neural networks for optimal deterministic universal quantum cloning. Phys. Rev. Research 5, 013139 (2023).](https://journals.aps.org/prresearch/pdf/10.1103/PhysRevResearch.5.013139)
11. [Gisin, N., Massar, S. (1997). Optimal quantum cloning machines. Phys. Rev. Lett. 79, 2153.](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.79.2153)
12. [Werner, R. F. (1998). Optimal cloning of pure states. Phys. Rev. A 58, 1827.](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.58.1827)
13. [High-fidelity quantum cloning of two nonorthogonal quantum states via weak measurements. Phys. Rev. A 99, 012324 (2019).](https://harvest.aps.org/v2/journals/articles/10.1103/PhysRevA.99.012324/fulltext)
14. [Kay, A., Ramanathan, R., Kaszlikowski, D. (2012). Optimal asymmetric quantum cloning. arXiv.](https://arxiv.org/abs/1208.5574)
15. [Distribution of quantum Fisher information in asymmetric cloning machines. PMC.](https://pmc.ncbi.nlm.nih.gov/articles/PMC4258659/)
16. [Duan, L.-M., Guo, G.-C. (1997). Linearly-independent quantum states can be cloned. arXiv.](https://arxiv.org/abs/quant-ph/9705018)
17. [Cloning of continuous quantum variables (full text via CiteSeerX)](https://citeseerx.ist.psu.edu/document?doi=8da2dc625da2d48d7cce3b350e849980e25d1212&repid=rep1&type=pdf)
18. [Cerf, N. J., Krüger, O., Navez, P., Werner, R. F., Wolf, M. M. (2005). Non-Gaussian cloning of quantum coherent states is optimal. Phys. Rev. Lett. 95, 070501.](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.95.070501)
19. [Huang, Y.-F., Li, W.-L., Li, C.-F., Zhang, Y.-S., Jiang, Y.-K., Guo, G.-C. (2001). Optical realization of universal quantum cloning. Phys. Rev. A 64, 012315.](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.64.012315)
20. [Lamas-Linares, A., Simon, C., Howell, J. C., Bouwmeester, D. (2002). Experimental quantum cloning of single photons. arXiv.](https://arxiv.org/abs/quant-ph/0205149)
21. [Quantum cloning. Rev. Mod. Phys. 77, 1225 (2005).](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.77.1225)
22. [Ferenczi, A., Lütkenhaus, N. (2012). Symmetries in quantum key distribution and the connection between optimal attacks and optimal cloning. Phys. Rev. A 85, 052310.](https://doi.org/10.1103/PhysRevA.85.052310)
23. [Quantum key distribution as a quantum machine learning task. npj Quantum Information (2025).](https://www.nature.com/articles/s41534-025-01088-9)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Quantum physics › Quantum information science › Quantum communication and information theory › Quantum communication primitives › No-cloning theorem*
*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*
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