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

Physical Review A

Optimal cloning of qubits from replicas of a qubit and its orthogonal states

Go Kato

Published Sep 16, 2010• 3 citations
Publisher

Abstract

We consider the situation where $s$ replicas of a qubit with an unknown state and its orthogonal $k$ replicas are given as an input, and we try to make $c$ clones of the qubit with the unknown state. As a function of $s$, $k$, and $c$, we obtain the optimal fidelity between the qubit with an unknown state and the clone by explicitly giving a completely positive trace-preserving (CPTP) map that represents a cloning machine. We discuss dependency of the fidelity on the values of the parameters $s$, $k$, and $c$.

DOI
10.1103/physreva.82.032314

Related papers

Cloning of qubits with both the cloned state and the state orthogonal to it as inputs

Go Kato

A universal cloning machine is derived. The machine produces $c$ clones of an unknown qubit from $s$ identical replicas of qubit and $k$ identical replicas of its orthogonal qubit. For the standard cloning machine, i.e., $k=0$, the universal-not machine, $s=0$, and some other cases, the optimum machine is well known. The universal cloning machine derived in this paper gives clones whose fidelity is the same value as that for the optimum machine. In the case of sufficiently large $c$, we show that the clone fidelity generated by the cloning machine from $s\ensuremath{\ne}0$ replicas and $k\ensuremath{\ne}0$ orthogonal replicas is higher than that generated by the optimal cloning machine from $s+k$ replicas in case $s+k\ensuremath{\le}300$.

Physical Review A•Mar 13, 2009•3 citations

Quantum cloning of orthogonal qubits

Jaromı́r Fiurášek, S. Iblisdir, Serge Massar +1 more

Abstract An optimal universal cloning transformation is derived that produces M copies of an unknown qubit from a pair of orthogonal qubits. For M>6, the corresponding cloning fidelity is higher than that of the optimal copying of a pair of identical qubits.

Fortschritte der Physik•Feb 26, 2003•1 citation

Quantum cloning of orthogonal qubits

Jaromı́r Fiurášek, S. Iblisdir, Serge Massar +1 more

An optimal universal cloning transformation is derived that produces M copies of an unknown qubit from a pair of orthogonal qubits. For $M>6,$ the corresponding cloning fidelity is higher than that of the optimal copying of a pair of identical qubits. It is shown that this cloning transformation can be implemented probabilistically via parametric down conversion by feeding the signal and idler modes of a nonlinear crystal with orthogonally polarized photons.

arXiv:quant-ph/0110016•Physical Review A•Apr 5, 2002•37 citations

Universal optimal cloning of qubits and quantum registers

Vladimír Bužek, Mark S. Hillery

We review our recent work on the universal (i.e. input state independent) optimal quantum copying (cloning) of qubits. We present unitary transformations which describe the optimal cloning of a qubit and we present the corresponding quantum logical network. We also present network for an optimal quantum copying ``machine'' (transformation) which produces N+1 identical copies from the original qubit. Here again the quality (fidelity) of the copies does not depend on the state of the original and is only a function of the number of copies, N. In addition, we present the machine which universaly and optimally clones states of quantum objects in arbitrary-dimensional Hilbert spaces. In particular, we discuss universal cloning of quantum registers.

arXiv:quant-ph/9801009•Jan 7, 1998•1 citation

Optimal cloning of mixed states

Marco Fanizza, Dmitry Grinko, Thilo Scharnhorst +1 more

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.

quant-pharXiv:2608.27298•Aug 28, 2026•0 citations

Optimal manipulations with qubits: universal quantum cloning and universal logical NOT operation

Vladimír Bužek, Mark S. Hillery

We analyze how information encoded in quantum systems can be optimally processed. In particular, we investigate copying (cloning) of quantum information (represented as states of 2-level quantum systems--qubits). We present unitary transformations which describe the optimal universal cloning of a qubit. Universality of the transformation guarantees that the fidelity of the cloning does not depend on the input state of the qubit, i.e. all states are cloned equally well. We present network for the optimal universal quantum cloning `machine' (transformation) which produces N + 1 copies from the original qubit. Here again the quality (fidelity) of the copies does not depend on the state of the original and is only a function of the number of copes, N. We also present the `machine' which universally and optimally clones states of quantum objects in arbitrary- dimensional Hilbert spaces. In particular, we discuss universal cloning of quantum registers. In addition to cloning of qubits we analyzed another universal operation-- the Universal NOT. We present the optimal transformation and the corresponding logical network which optimally complements an arbitrary input state of a qubit. We show that the fidelity of the performance of the Universal NOT operation increases as a function of the number of input qubits prepared in the same state.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Jul 14, 1999•0 citations
We consider the situation where sss replicas of a qubit with an unknown state and its orthogonal kkk replicas are given as an input, and we try to make ccc clones of the qubit with the unknown state. As a function of sss, kkk, and ccc, we obtain the optimal fidelity between the qubit with an unknown state and the clone by explicitly giving a completely positive trace-preserving (CPTP) map that represents a cloning machine. We discuss dependency of the fidelity on the values of the parameters sss, kkk, and ccc.
A universal cloning machine is derived. The machine produces ccc clones of an unknown qubit from sss identical replicas of qubit and kkk identical replicas of its orthogonal qubit. For the standard cloning machine, i.e., k=0k=0k=0, the universal-not machine, s=0s=0s=0, and some other cases, the optimum machine is well known. The universal cloning machine derived in this paper gives clones whose fidelity is the same value as that for the optimum machine. In the case of sufficiently large ccc, we show that the clone fidelity generated by the cloning machine from $s\ensuremath{\ne}0$ replicas and $k\ensuremath{\ne}0$ orthogonal replicas is higher than that generated by the optimal cloning machine from s+ks+ks+k replicas in case $s+k\ensuremath{\le}300$.
An optimal universal cloning transformation is derived that produces M copies of an unknown qubit from a pair of orthogonal qubits. For M>6,M>6,M>6, the corresponding cloning fidelity is higher than that of the optimal copying of a pair of identical qubits. It is shown that this cloning transformation can be implemented probabilistically via parametric down conversion by feeding the signal and idler modes of a nonlinear crystal with orthogonally polarized photons.
We review our recent work on the universal (i.e. input state independent) optimal quantum copying (cloning) of qubits. We present unitary transformations which describe the optimal cloning of a qubit and we present the corresponding quantum logical network. We also present network for an optimal quantum copying ``machine'' (transformation) which produces N+1 identical copies from the original qubit. Here again the quality (fidelity) of the copies does not depend on the state of the original and is only a function of the number of copies, N. In addition, we present the machine which universaly and optimally clones states of quantum objects in arbitrary-dimensional Hilbert spaces. In particular, we discuss universal cloning of quantum registers.
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.
We analyze how information encoded in quantum systems can be optimally processed. In particular, we investigate copying (cloning) of quantum information (represented as states of 2-level quantum systems--qubits). We present unitary transformations which describe the optimal universal cloning of a qubit. Universality of the transformation guarantees that the fidelity of the cloning does not depend on the input state of the qubit, i.e. all states are cloned equally well. We present network for the optimal universal quantum cloning `machine' (transformation) which produces N + 1 copies from the original qubit. Here again the quality (fidelity) of the copies does not depend on the state of the original and is only a function of the number of copes, N. We also present the `machine' which universally and optimally clones states of quantum objects in arbitrary- dimensional Hilbert spaces. In particular, we discuss universal cloning of quantum registers. In addition to cloning of qubits we analyzed another universal operation-- the Universal NOT. We present the optimal transformation and the corresponding logical network which optimally complements an arbitrary input state of a qubit. We show that the fidelity of the performance of the Universal NOT operation increases as a function of the number of input qubits prepared in the same state.