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:quant-ph/0101100 · Physical Review A

Optimal nonuniversally covariant cloning

Giacomo Mauro D’Ariano, Paoloplacido Lo Presti

Published Sep 12, 2001• 99 citations
arXivHTMLPDF

Abstract

We consider nonuniversal cloning maps, namely, cloning transformations that are covariant under a proper subgroup G of the universal unitary group $U(d),$ where d is the dimension of the Hilbert space $\mathcal{H}$ of the system to be cloned. We give a general method for optimizing cloning for any cost function. Examples of applications are given for phase-covariant cloning (cloning of equatorial qubits and qutrits) and for the Weyl-Heisenberg group (cloning of ``continuous variables'').

DOI
10.1103/physreva.64.042308

Related papers

Optimal phase-covariant cloning for qubits and qutrits

Giacomo Mauro D’Ariano, Chiara Macchiavello

We consider cloning transformations of equatorial qubits $|{\ensuremath{\psi}}_{\ensuremath{\varphi}}〉=1/\sqrt{2}(|0〉{+e}^{i\ensuremath{\varphi}}|1〉)$ and qutrits $|{\ensuremath{\psi}}_{\ensuremath{\varphi},\ensuremath{\theta}}〉=1/\sqrt{3}(|0〉{+e}^{i\ensuremath{\varphi}}|1〉{+e}^{i\ensuremath{\theta}}|2〉),$ with the transformation covariant for rotation of the phases $\ensuremath{\varphi}$ and $\ensuremath{\theta}.$ The optimal cloning maps are derived without simplifying assumptions from first principles, for any number of input and output qubits, and for a single-input qutrit and any number of output qutrits. We also compare the cloning maps for global and single-particle fidelities, and we show that the two criteria lead to different optimal maps.

arXiv:quant-ph/0301175•Physical Review A•Apr 11, 2003•104 citations

Asymmetric phase covariant d-dimensional cloning

L.-P. Lamoureux, Nicolas J. Cerf

We consider cloning transformations of d-dimensional states of the form $e^{i\phi_0}|0> + e^{i\phi_1}|1> +...+ e^{i\phi_{d-1}}|d-1>$ that are covariant with respect to rotations of the phases $\phi_i$'s. The optimal cloning maps are easily obtained using a well-defined general characterization of state-dependent $1 \rightarrow 2$ cloning transformations in arbitrary dimensions. Our results apply to symmetric as well as asymmetric cloners, so that the balance between the fidelity of the two clones can be analyzed.

arXiv:quant-ph/0410054•Quantum Information and Computation•Jan 1, 2005•8 citations

Optimal Phase-Covariant Cloning in 3 Dimensions

Wen‐Hai Zhang, Longbao Yu, Zhuo‐Liang Cao +1 more

In this paper, we present the explicit transformations of the optimal 1 → 3, 4, 5 phase-covariant cloning in 3 dimensions. The cloning fidelities are covered by the theoretical bounds of the optimal 1 → 3k, 3k + 1, 3k + 2 phase-covariant cloning of qutrits, where k ≥ 1 is the integral [Phys. Rev. A 67 (2003) 042306].

Communications in Theoretical Physics•Jun 1, 2012•0 citations

General phase-covariant quantum cloning

Lan-Lan Li, Wen‐Hai Zhang, Ke Zhang +1 more

Phase-covariant quantum cloning has important applications in quantum cryptography. This paper proposes the unitary transformation of the optimal 1 → 2 general phase covariant quantum cloning in two dimensions and obtains the maximal fidelity of copies. In addition, an experimental scheme is designed to implement this unitary transformation using linear optical elements, and the scheme is feasible under the present experimental technology.

AIP Advances•Dec 1, 2024•0 citations

Economical realization of phase-covariant devices in arbitrary dimensions (Invited)

Francesco Buscemi, Giacomo Mauro D’Ariano, Chiara Macchiavello

We describe a unified framework of phase-covariant multiuser quantum transformations for d-dimensional quantum systems. We derive the optimal phase-covariant cloning and transposition transformations for multiphase states. We show that for some particular relations between the input and output number of copies, they correspond to economical transformations, which can be achieved without the need of auxiliary systems. We prove a relation between the optimal phase-covariant cloning and transposition maps and optimal estimation of multiple phases for equatorial states.

arXiv:quant-ph/0606120•Journal of the Optical Society of America B•Jan 26, 2007•5 citations

Extremal quantum cloning machines

Giulio Chiribella, Giacomo Mauro D’Ariano, Paolo Perinotti +1 more

We investigate the problem of cloning a set of states that is invariant under the action of an irreducible group representation. We then characterize the cloners that are extremal in the convex set of group covariant cloning machines, among which one can restrict the search for optimal cloners. For a set of states that is invariant under the discrete Weyl-Heisenberg group, we show that all extremal cloners can be unitarily realized using the so-called double-Bell states, whence providing a general proof of the popular ansatz used in the literature for finding optimal cloners in a variety of settings. Our result can also be generalized to continuous-variable optimal cloning in infinite dimensions, where the covariance group is the customary Weyl-Heisenberg group of displacements.

arXiv:quant-ph/0507130•Physical Review A•Oct 28, 2005•23 citations
We consider nonuniversal cloning maps, namely, cloning transformations that are covariant under a proper subgroup G of the universal unitary group U(d),U(d),U(d), where d is the dimension of the Hilbert space H\mathcal{H}H of the system to be cloned. We give a general method for optimizing cloning for any cost function. Examples of applications are given for phase-covariant cloning (cloning of equatorial qubits and qutrits) and for the Weyl-Heisenberg group (cloning of ``continuous variables'').
We consider cloning transformations of equatorial qubits $|{\ensuremath{\psi}}_{\ensuremath{\varphi}}〉=1/\sqrt{2}(|0〉{+e}^{i\ensuremath{\varphi}}|1〉)$ and qutrits $|{\ensuremath{\psi}}_{\ensuremath{\varphi},\ensuremath{\theta}}〉=1/\sqrt{3}(|0〉{+e}^{i\ensuremath{\varphi}}|1〉{+e}^{i\ensuremath{\theta}}|2〉),$ with the transformation covariant for rotation of the phases $\ensuremath{\varphi}$ and $\ensuremath{\theta}.$ The optimal cloning maps are derived without simplifying assumptions from first principles, for any number of input and output qubits, and for a single-input qutrit and any number of output qutrits. We also compare the cloning maps for global and single-particle fidelities, and we show that the two criteria lead to different optimal maps.
We consider cloning transformations of d-dimensional states of the form eiϕ0∣0>+eiϕ1∣1>+...+eiϕd−1∣d−1>e^{i\phi_0}|0> + e^{i\phi_1}|1> +...+ e^{i\phi_{d-1}}|d-1>eiϕ0​∣0>+eiϕ1​∣1>+...+eiϕd−1​∣d−1> that are covariant with respect to rotations of the phases ϕi\phi_iϕi​'s. The optimal cloning maps are easily obtained using a well-defined general characterization of state-dependent 1→21 \rightarrow 21→2 cloning transformations in arbitrary dimensions. Our results apply to symmetric as well as asymmetric cloners, so that the balance between the fidelity of the two clones can be analyzed.
We describe a unified framework of phase-covariant multiuser quantum transformations for d-dimensional quantum systems. We derive the optimal phase-covariant cloning and transposition transformations for multiphase states. We show that for some particular relations between the input and output number of copies, they correspond to economical transformations, which can be achieved without the need of auxiliary systems. We prove a relation between the optimal phase-covariant cloning and transposition maps and optimal estimation of multiple phases for equatorial states.
We investigate the problem of cloning a set of states that is invariant under the action of an irreducible group representation. We then characterize the cloners that are extremal in the convex set of group covariant cloning machines, among which one can restrict the search for optimal cloners. For a set of states that is invariant under the discrete Weyl-Heisenberg group, we show that all extremal cloners can be unitarily realized using the so-called double-Bell states, whence providing a general proof of the popular ansatz used in the literature for finding optimal cloners in a variety of settings. Our result can also be generalized to continuous-variable optimal cloning in infinite dimensions, where the covariance group is the customary Weyl-Heisenberg group of displacements.