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arXiv:quant-ph/0411179 · Physical Review A

Multipartite asymmetric quantum cloning

S. Iblisdir, Antonio Acín, Nicolas J. Cerf, Radim Filip, Jaromı́r Fiurášek, N. Gisin

Published Oct 26, 2005• 70 citations
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Abstract

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.

DOI
10.1103/physreva.72.042328

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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→MA+MBN\ensuremath{\rightarrow}{M}_{A}+{M}_{B}N\ensuremath→MA​+MB​ cloning machines are exhibited. We also present an optimal 1\ensuremath→1+1+11\ensuremath{\rightarrow}1+1+11\ensuremath→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.
We study machines that take N identical replicas of a pure qudit state as input and output a set of M_A clones of a given fidelity and another set of MBM_BMB​ clones of another fidelity. The trade-off between these two fidelities is investigated, and numerous examples of optimal N -> M_A+M_B cloning machines are exhibited using a generic method. A generalisation to more than two sets of clones is also discussed. Finally, an optical implementation of some such machines is proposed. This paper is an extended version of [xxx.arxiv.org/abs/quant-ph/0411179].
We study machines that take N identical replicas of a pure qudit state as input and output a set of M_A clones of a given fidelity and another set of M_B clones of another fidelity. The trade-off between these two fidelities is investigated, and numerous examples of optimal N \to M_A+M_B cloning machines are exhibited using a generic method. A generalisation to more than two sets of clones is also discussed. Finally, an optical implementation of some such machines is proposed. This paper is an extended version of Phys. Rev. A 72, 042328 (2005).
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.
We investigate the universal asymmetric cloning of states in a Hilbert space of arbitrary dimension. We derive the class of optimal and fully asymmetric 1→31\rightarrow 31→3 cloners, which produce three copies, each having a different fidelity. A simple parametric expression for the maximum achievable cloning fidelity triplets is then provided. As a side-product, we also prove the optimality of the 1→21\rightarrow 21→2 asymmetric cloning machines that have been proposed in the literature.
An unknown quantum state cannot be copied and broadcast freely due to the no-cloning theorem. Approximate cloning schemes have been proposed to achieve the optimal cloning characterized by the maximal fidelity between the original and its copies. Here, from the perspective of quantum Fisher information (QFI), we investigate the distribution of QFI in asymmetric cloning machines which produce two nonidentical copies. As one might expect, improving the QFI of one copy results in decreasing the QFI of the other copy. It is perhaps also unsurprising that asymmetric phase-covariant cloning outperforms universal cloning in distributing QFI since a priori information of the input state has been utilized. However, interesting results appear when we compare the distributabilities of fidelity (which quantifies the full information of quantum states), and QFI (which only captures the information of relevant parameters) in asymmetric cloning machines. Unlike the results of fidelity, where the distributability of symmetric cloning is always optimal for any d-dimensional cloning, we find that any asymmetric cloning outperforms symmetric cloning on the distribution of QFI for d ≤ 18, whereas some but not all asymmetric cloning strategies could be worse than symmetric ones when d > 18.