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arXiv:0807.1657 · Physical Review A

Role of interference in quantum cloning

Benoit Roubert, Daniel Braun

Published Oct 10, 2008• 6 citations
arXivHTMLPDF

Abstract

We study the role of interference in the process of quantum cloning. We show that in order to achieve better than classical cloning of a qubit no interference is needed. In particular, a large class of symmetric universal $1\ensuremath{\rightarrow}2$ qubit cloners exists which achieve the optimal average fidelity $5∕6$ for such machines, without using any interference in the propagation of the original and copied qubits. We also obtain optimal average fidelities for interference-free cloning in asymmetric situations, and discuss the relation of the quantum cloners found to the Bu\ifmmode \check{z}\else \v{z}\fi{}ek-Hillery quantum cloner.

DOI
10.1103/physreva.78.042311

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We study the role of interference in the process of quantum cloning. We show that in order to achieve better than classical cloning of a qubit no interference is needed. In particular, a large class of symmetric universal 1\ensuremath→21\ensuremath{\rightarrow}21\ensuremath→2 qubit cloners exists which achieve the optimal average fidelity 5∕65∕65∕6 for such machines, without using any interference in the propagation of the original and copied qubits. We also obtain optimal average fidelities for interference-free cloning in asymmetric situations, and discuss the relation of the quantum cloners found to the Bu\ifmmode \check{z}\else \v{z}\fi{}ek-Hillery quantum cloner.
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 56\frac{5}{6}65​ (which proved to be the optimal fidelity of $\stackrel{\ensuremath{\rightarrow}}{1}$2 qubit universal cloner) for arbitrary input pure states.
We establish the best possible approximation to a perfect quantum cloning machine that produces two clones out of a single input. We analyze both universal and state-dependent cloners. The maximal fidelity of cloning is shown to be 5/6 for universal cloners. It can be achieved either by a special unitary evolution or by a teleportation scheme. We construct the optimal state-dependent cloners operating on any prescribed two nonorthogonal states and discuss their fidelities and the use of auxiliary physical resources in the process of cloning. The optimal universal cloners permit us to derive an upper bound on the quantum capacity of the depolarizing quantum channel.
After introducing the no-cloning theorem and the most common forms of approximate quantum cloning, universal quantum cloning is considered in detail. The connections it has with universal NOT-gate, quantum cryptography and state estimation are presented and briefly discussed. The state estimation connection is used to show that the amount of extractable classical information and total Bloch vector length are conserved in universal quantum cloning. The 1  2 qubit cloner is also shown to obey a complementarity relation between local and nonlocal information. These are interpreted to be a consequence of the conservation of total information in cloning. Finally, the performance of the 1  M cloning network discovered by Bužek, Hillery and Knight is studied in the presence of decoherence using the Barenco et al. approach where random phase fluctuations are attached to 2-qubit gates. The expression for average fidelity is calculated for three cases and it is found to depend on the optimal fidelity and the average of the phase fluctuations in a specific way. It is conjectured to be the form of the average fidelity in the general case. While the cloning network is found to be rather robust, it is nevertheless argued that the scalability of the quantum network implementation is poor by studying the effect of decoherence during the preparation of the initial state of the cloning machine in the 1 ! 2 case and observing that the loss in average fidelity can be large. This affirms the result by Maruyama and Knight, who reached the same conclusion in a slightly different manner.
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 present quantum cloning machines that transform NNN identical qubits into M>NM>NM>N identical copies and we prove that the fidelity (quality) of these copies is optimal. The connection between cloning and measurement is discussed in detail. When the number of clones MMM tends towards infinity, the fidelity of each clone tends towards the optimal fidelity that can be obtained by a measurement on the input qubits. More generally, quantum cloning machines are universal devices to translate quantum information into classical information.