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arXiv:quant-ph/0006032 · Physical Review A

Optical realization of universal quantum cloning

Yun‐Feng Huang, Wanli Li, Chuan‐Feng Li, Yong-Sheng Zhang, Yun-Kun Jiang, Guang‐Can Guo

Published Jun 13, 2001• 49 citations
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Abstract

Beyond the no-cloning theorem, the universal symmetric quantum cloning machine was first addressed by Bu\ifmmode \check{z}\else \v{z}\fi{}ek and Hillery. Here, we realized the $\stackrel{\ensuremath{\rightarrow}}{1}$2 qubit Bu\ifmmode \check{z}\else \v{z}\fi{}ek-Hillery cloning machine with linear optical devices. This method relies on the representation of several qubits by a single photon. We showed that the fidelities between the two output qubits and the original qubit are both $\frac{5}{6}$ (which proved to be the optimal fidelity of $\stackrel{\ensuremath{\rightarrow}}{1}$2 qubit universal cloner) for arbitrary input pure states.

DOI
10.1103/physreva.64.012315

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Beyond the no-cloning theorem, the universal symmetric quantum cloning machine was first addressed by Bu\ifmmode \check{z}\else \v{z}\fi{}ek and Hillery. Here, we realized the 1\ensuremath→\stackrel{\ensuremath{\rightarrow}}{1}1\ensuremath→​2 qubit Bu\ifmmode \check{z}\else \v{z}\fi{}ek-Hillery cloning machine with linear optical devices. This method relies on the representation of several qubits by a single photon. We showed that the fidelities between the two output qubits and the original qubit are both 56\frac{5}{6}65​ (which proved to be the optimal fidelity of 1\ensuremath→\stackrel{\ensuremath{\rightarrow}}{1}1\ensuremath→​2 qubit universal cloner) for arbitrary input pure states.
We propose an experimentally unified linear optical scheme to implement the optimal symmetric one to two (1 → 2) universal, optimal symmetric 1 → 2 phase-covariant and optimal symmetric economical one to three (1 → 3) phase-covariant quantum cloning machines of the polarization state of the single photon. This scheme requires single-photon sources as the input state. Here, we use a laser beam attenuated to the single-photon level as the input source. The scheme relies on one polarized qubit and two location qubits and it also involves linear optical elements. It is shown that under certain conditions, the scheme is feasible by current experimental technology.
We present a unified universal quantum cloning machine, which combines several different existing universal cloning machines together, including the asymmetric case. In this unified framework, the identical pure states are projected equally into each copy initially constituted by input and one half of the maximally entangled states. We show explicitly that the output states of those universal cloning machines are the same. One importance of this unified cloning machine is that the cloning procession is always the symmetric projection, which reduces dramatically the difficulties for implementation. Also, it is found that this unified cloning machine can be directly modified to the general asymmetric case. Besides the global fidelity and the single-copy fidelity, we also present all possible arbitrary-copy fidelities.
We show that optimal universal cloning of the polarization state of photons can be achieved via stimulated emission in three-level systems, both of the \ensuremath{\Lambda} and the VVV type. We establish the equivalence of our systems with coupled harmonic oscillators, which permits us to analyze the structure of the cloning transformations realized. These transformations are shown to be equivalent to the optimal cloning transformations for qubits discovered by Bu\ifmmode \check{z}\else \v{z}\fi{}ek and Hillery and Gisin and Massar. The down-conversion cloner discovered previously by some of the authors is obtained as a limiting case. We demonstrate an interesting equivalence between systems of \ensuremath{\Lambda} atoms and systems of pairwise entangled V atoms. Finally we discuss the physical differences between our photon cloners and the qubit cloners considered previously and prove that the bounds on the fidelity of the clones derived for qubits also apply in our situation.
After a brief introduction to the quantum no-cloning theorem and its link with the linearity and causality of quantum mechanics, the concept of quantum cloning machines is sketched, following, whenever possible, the chronology of the main results. The important classes of quantum cloning machines are reviewed, in particular state-independent and state-dependent cloning machines. The 1-to-2 cloning problem is then studied from a formal point of view, using the isomorphism between completely positive maps and operators, which leads to the so-called double-Bell ansatz. This also yields an efficient numerical approach to quantum cloning, based on semidefinite programming methods. The derivation of the optimal N-to-M universal cloning machine in d dimensions is then detailed, as well as the notion of asymmetric cloning machines. In the second part of this review, the optical implementation of cloning machines is considered. It is shown that the universal cloning of photons can be achieved by parametric amplification of light or by symmetrization via the Hong-Ou-Mandel effect. The various experimental demonstrations of quantum cloning machines are reviewed. The cloning of orthogonally polarized photons is also considered, as well as the asymmetric and phase-covariant cloning of photons. Finally, the extension of quantum cloning to continuous variables is analyzed. The optimal cloning of coherent states of light by phase-insensitive amplification is explained, as well as the experimental realization of continuous-variable quantum cloning with linear optics, measurement, and feed-forward operations.
Due to the axioms of quantum mechanics, perfect cloning of an unknown quantum state is impossible. But since imperfect cloning is still possible, a question arises: ``Is there an optimal quantum cloning machine?'' Buzek and Hillery answered this question and constructed their famous B-H quantum cloning machine. The B-H machine clones the state of an arbitrary single qubit in an optimal manner and hence it is universal. Generalizing this machine for a two-qubit system is straightforward, but during this procedure, except for product states, this machine loses its universality and becomes a state-dependent cloning machine. In this paper, we propose some classes of optimal universal local quantum state cloners for a particular class of two-qubit systems, more precisely, for a class of states with known Schmidt basis. We then extend our machine to the case that the Schmidt basis of the input state is deviated from the local computational basis of the machine. We show that more local quantum coherence existing in the input state corresponds to less fidelity between the input and output states. Also we present two classes of a state-dependent local quantum copying machine. Furthermore, we investigate local broadcasting of two aspects of quantum correlations, i.e., quantum entanglement and quantum discord, defined, respectively, within the entanglement-separability paradigm and from an information-theoretic perspective. The results show that although quantum correlation is, in general, very fragile during the broadcasting procedure, quantum discord is broadcasted more robustly than quantum entanglement.