We derive a tight upper bound for the fidelity of a universal $N\ensuremath{\rightarrow}M$ qubit cloner, valid for any $M\ensuremath{\ge}N$, where the output of the cloner is required to be supported on the symmetric subspace. Our proof is based on the concatenation of two cloners and the connection between quantum cloning and quantum state estimation. We generalize the operation of a quantum cloner to mixed and/or entangled input qubits described by a density matrix supported on the symmetric subspace of the constituent qubits. We also extend the validity of optimal state estimation methods to inputs of this kind.We derive a tight upper bound for the fidelity of a universal $N\ensuremath{\rightarrow}M$ qubit cloner, valid for any $M\ensuremath{\ge}N$, where the output of the cloner is required to be supported on the symmetric subspace. Our proof is based on the concatenation of two cloners and the connection between quantum cloning and quantum state estimation. We generalize the operation of a quantum cloner to mixed and/or entangled input qubits described by a density matrix supported on the symmetric subspace of the constituent qubits. We also extend the validity of optimal state estimation methods to inputs of this kind.

arXiv:quant-ph/9712019Physical Review LettersSep 21, 1998283 citations

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.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:2608.27298Aug 28, 20260 citations

Optimal Universal Cloning of Mixed States

Gui-Fang Dang, Heng Fan

Sep 23, 20060 citations

Optimal Universal Quantum Cloning and State Estimation

Dagmar Bruß, ARTUR K. EKERT, Chiara Macchiavello

WORLD SCIENTIFIC eBooksFeb 1, 20050 citations

Analytical expression of the geometric quantum discord generated by asymmetric cloning

Irina Ion, Iulia Ghiu

We analyze the output mixed states generated when the asymmetric cloners are applied on a pure two-qubit state. We investigate two kinds of cloners: (a) local [Formula: see text] optimal universal asymmetric cloning machines and (b) nonlocal [Formula: see text] optimal universal asymmetric cloning machines. In both cases we find analytical formulae of the geometric quantum discord of the two clones. Finally, we prove that the geometric quantum discord generated by the nonlocal asymmetric cloners is not smaller than the geometric quantum discord given by the local asymmetric cloners.We analyze the output mixed states generated when the asymmetric cloners are applied on a pure two-qubit state. We investigate two kinds of cloners: (a) local [Formula: see text] optimal universal asymmetric cloning machines and (b) nonlocal [Formula: see text] optimal universal asymmetric cloning machines. In both cases we find analytical formulae of the geometric quantum discord of the two clones. Finally, we prove that the geometric quantum discord generated by the nonlocal asymmetric cloners is not smaller than the geometric quantum discord given by the local asymmetric cloners.

International Journal of Quantum InformationDec 30, 20250 citations

Optimal Purification of Arbitrary Quantum Mixed States

Yuan Ping, Huina Li, Xiuqin Pan +2 more

International Journal of Theoretical PhysicsAug 8, 20134 citations

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.