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Journal of Modern Optics

Optimal quantum cloning and universal NOT without quantum gates

Christoph Simon, Gregor Weihs, Anton Zeilinger

Published Feb 1, 2000• 15 citations
Publisher

Abstract

We present ways of realizing quantum cloning via stimulated emission. Universality of the cloning procedure is achieved by choosing systems that have appropriate symmetries. We first discuss a scheme based on certain three-level systems, e.g. atoms in a cavity. Our numerical results show that this scheme approaches optimal cloning for short interaction times. Then we demonstrate that optimal universal cloning can be realized using parametric down-conversion. At the same time, our down-conversion scheme also implements the optimal universal NOT operation. We conclude with some remarks on cloning and superluminal signalling, using our cloner as an illustrative example.

DOI
10.1080/09500340008244038

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We present ways of realizing quantum cloning via stimulated emission. Universality of the cloning procedure is achieved by choosing systems that have appropriate symmetries. We first discuss a scheme based on certain three-level systems, e.g. atoms in a cavity. Our numerical results show that this scheme approaches optimal cloning for short interaction times. Then we demonstrate that optimal universal cloning can be realized using parametric down-conversion. At the same time, our down-conversion scheme also implements the optimal universal NOT operation. We conclude with some remarks on cloning and superluminal signalling, using our cloner as an illustrative example.
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.
Although perfect copying of unknown quantum systems is forbidden by the laws of quantum mechanics, approximate cloning is possible. A natural way of realizing quantum cloning of photons is by stimulated emission. In this context, the fundamental quantum limit to the quality of the clones is imposed by the unavoidable presence of spontaneous emission. In our experiment, a single input photon stimulates the emission of additional photons from a source on the basis of parametric down-conversion. This leads to the production of quantum clones with near-optimal fidelity. We also demonstrate universality of the copying procedure by showing that the same fidelity is achieved for arbitrary input states.
This thesis consists of four parts. In the first part it is shown that optimal universal cloning of photons can be realized with the help of stimulated emission. Possible schemes based on three-level systems and on parametric down-conversion are analyzed in detail. The second part shows that the dynamical rules of quantum physics can be derived from its static properties and the condition that superluminal communication is forbidden. In the third part a very simple form of the Kochen-Specker theorem is presented, which leads to a proposal for a simple experimental test of non-contextual hidden variables. The fourth part shows how hidden-variable theorems can be derived in a completely operational way. In particular a Kochen-Specker theorem is derived that applies to experiments with finite precision.
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.
We present a scheme to realize a special quantum cloning machine via input-output cavities. The cloning machine can copy information from one atom to another distant atom. Choosing different parameters, the method can perform optimal symmetric (asymmetric) universal quantum cloning and optimal symmetric (asymmetric) phase-covariant cloning. Compared to previous schemes, our scheme is more insensitive to actual environmental noise and can get higher fidelity in a current cavity quantum electrodynamics system with an entangled state acting as a quantum channel than a single-photon pulse.