Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1407.4951 · Quantum Information and Computation

Optimal universal quantum cloning: asymmetries and fidelity measures

Alastair Kay

Published Sep 1, 2016• 10 citations
arXivHTMLPDF

Abstract

We study the problem of universal quantum cloning – taking several identical copies of a pure but unknown quantum state and producing further copies. While it is well known that it is impossible to perfectly reproduce the state, how well the copies can be cloned can be quantified using the fidelity. We examine how individual fidelities can be traded against each other, and how different fidelity measures can be incorporated. The broadly applicable formalism into which we transform the cloning problem is described as a series of quadratic constraints which are amenable to mathematical and computational scrutiny. As such, we reproduce all known results on optimal universal cloning, and push the recent results on asymmetric cloning much further, giving new trade-off relations between fidelities for broad classes of optimal cloning machines. We also provide substantial evidence that motivates why other parameter ranges (number of input copies) have not, and will not, yield to similar analysis.

DOI
10.26421/qic16.11-12-5

Related papers

Fidelity Measures in Asymmetric Quantum Cloning

Alastair Kay

We study the problem of universal quantum cloning -- taking several identical copies of a pure but unknown quantum state and producing further copies. While it is well known that it is impossible to perfectly reproduce the state, how well the copies can be cloned can be quantified using the fidelity. We examine how individual fidelities can be traded against each other, and how different fidelity measures can be incorporated. The broadly applicable formalism into which we transform the cloning problem is described as a series of quadratic constraints which are amenable to mathematical and computational scrutiny. As such, we reproduce all known results on optimal universal cloning, and push the recent results on asymmetric cloning much further, giving new trade-off relations between fidelities for broad classes of optimal cloning machines. We also provide substantial evidence that motivates why other parameter ranges (number of input copies) have not, and will not, yield to similar analysis.

•Jul 18, 2014•0 citations

Optimal universal quantum cloning

KayAlastair

We study the problem of universal quantum cloning - taking several identical copies of a pure but unknown quantum state and producing further copies. While it is well known that it is impossible to...

Quantum Information and Computation•Sep 1, 2016•0 citations

Optimal Asymmetric Quantum Cloning

Alastair Kay, Ravishankar Ramanathan, Dagomir Kaszlikowski

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.

arXiv:1208.5574•Aug 28, 2012•3 citations

Multipartite asymmetric quantum cloning

S. Iblisdir, Antonio Acín, Nicolas J. Cerf +3 more

We investigate the optimal distribution of quantum information over multipartite systems in asymmetric settings. We introduce cloning transformations that take $N$ identical replicas of a pure state in any dimension as input and yield a collection of clones with nonidentical fidelities. As an example, if the clones are partitioned into a set of ${M}_{A}$ clones with fidelity ${F}^{A}$ and another set of ${M}_{B}$ clones with fidelity ${F}^{B}$, the trade-off between these fidelities is analyzed, and particular cases of optimal $N\ensuremath{\rightarrow}{M}_{A}+{M}_{B}$ cloning machines are exhibited. We also present an optimal $1\ensuremath{\rightarrow}1+1+1$ cloning machine, which is an example of a tripartite fully asymmetric cloner. Finally, it is shown how these cloning machines can be optically realized.

arXiv:quant-ph/0411179•Physical Review A•Oct 26, 2005•70 citations

Optimal copy complexity of quantum state cloning

Sangwoo Jeon, Vaughn Sohn, Changhun Oh

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.

quant-pharXiv:2608.24484•Aug 26, 2026•0 citations

Optimal asymmetric quantum cloning for quantum information and computation

Alastair Kay, Ravishankar Ramanathan, Dagomir Kaszlikowski

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 $M\rightarrow N$ asymmetric cloning of qudits are obtained in a number of interesting cases. The central result is the solution to the $1 \rightarrow N$ 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 $1 \rightarrow N$ 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 Information and Computation•Sep 1, 2013•8 citations
We study the problem of universal quantum cloning – taking several identical copies of a pure but unknown quantum state and producing further copies. While it is well known that it is impossible to perfectly reproduce the state, how well the copies can be cloned can be quantified using the fidelity. We examine how individual fidelities can be traded against each other, and how different fidelity measures can be incorporated. The broadly applicable formalism into which we transform the cloning problem is described as a series of quadratic constraints which are amenable to mathematical and computational scrutiny. As such, we reproduce all known results on optimal universal cloning, and push the recent results on asymmetric cloning much further, giving new trade-off relations between fidelities for broad classes of optimal cloning machines. We also provide substantial evidence that motivates why other parameter ranges (number of input copies) have not, and will not, yield to similar analysis.
We study the problem of universal quantum cloning -- taking several identical copies of a pure but unknown quantum state and producing further copies. While it is well known that it is impossible to perfectly reproduce the state, how well the copies can be cloned can be quantified using the fidelity. We examine how individual fidelities can be traded against each other, and how different fidelity measures can be incorporated. The broadly applicable formalism into which we transform the cloning problem is described as a series of quadratic constraints which are amenable to mathematical and computational scrutiny. As such, we reproduce all known results on optimal universal cloning, and push the recent results on asymmetric cloning much further, giving new trade-off relations between fidelities for broad classes of optimal cloning machines. We also provide substantial evidence that motivates why other parameter ranges (number of input copies) have not, and will not, yield to similar analysis.
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.
We investigate the optimal distribution of quantum information over multipartite systems in asymmetric settings. We introduce cloning transformations that take NNN identical replicas of a pure state in any dimension as input and yield a collection of clones with nonidentical fidelities. As an example, if the clones are partitioned into a set of MA{M}_{A}MA​ clones with fidelity FA{F}^{A}FA and another set of MB{M}_{B}MB​ clones with fidelity FB{F}^{B}FB, the trade-off between these fidelities is analyzed, and particular cases of optimal $N\ensuremath{\rightarrow}{M}_{A}+{M}_{B}$ cloning machines are exhibited. We also present an optimal $1\ensuremath{\rightarrow}1+1+1$ cloning machine, which is an example of a tripartite fully asymmetric cloner. Finally, it is shown how these cloning machines can be optically realized.
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→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.