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Physical Review A

Optical implementations of the optimal phase-covariant quantum cloning machine

Jaromı́r Fiurášek

Published May 28, 2003• 101 citations
Publisher

Abstract

We propose two simple implementations of the optimal symmetric $\stackrel{\ensuremath{\rightarrow}}{1}2$ phase-covariant cloning machine for qubits. The first scheme is designed for qubits encoded into polarization states of photons and it involves a mixing of two photons on an unbalanced beam splitter. This scheme is probabilistic and the cloning succeeds with the probability $1/3.$ In the second setup, the qubits are represented by the states of Rydberg atoms and the cloning is accomplished by the resonant interaction of the atoms with a microwave field confined in a high-Q cavity. This latter approach allows for deterministic implementation of the optimal cloning transformation.

DOI
10.1103/physreva.67.052314

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We propose two simple implementations of the optimal symmetric 1\ensuremath→2\stackrel{\ensuremath{\rightarrow}}{1}21\ensuremath→​2 phase-covariant cloning machine for qubits. The first scheme is designed for qubits encoded into polarization states of photons and it involves a mixing of two photons on an unbalanced beam splitter. This scheme is probabilistic and the cloning succeeds with the probability 1/3.1/3.1/3. In the second setup, the qubits are represented by the states of Rydberg atoms and the cloning is accomplished by the resonant interaction of the atoms with a microwave field confined in a high-Q cavity. This latter approach allows for deterministic implementation of the optimal cloning transformation.
Quantum cloning relates to the security of quantum computation and quantum communication. In this paper, firstly we propose a feasible unified scheme to implement optimal 1 → 2 universal, 1 → 2 asymmetric and symmetric phase-covariant cloning, and 1 → 2 economical phase-covariant quantum cloning machines only via a beam splitter. Then 1 → 3 economical phase-covariant quantum cloning machines also can be realized by adding another beam splitter in context of linear optics. The scheme is based on the interference of two photons on a beam splitter with different splitting ratios for vertical and horizontal polarization components. It is shown that under certain condition, the scheme is feasible by current experimental technology.
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.
The experimental realization of optimal symmetric phase-covariant $1\ensuremath{\rightarrow}2$ cloning of qubit states is presented. The qubits are represented by polarization states of photons generated by spontaneous parametric down-conversion. The experiment is based on the interference of two photons on a custom-made beam splitter with different splitting ratios for vertical and horizontal polarization components. From the measured data we have estimated the implemented cloning transformation using the maximum-likelihood method. The result shows that the realized transformation is very close to the ideal one and the map fidelity reaches 94%.
We compare several optical implementations of phase-covariant cloning machines. The experiments are based on copying of the polarization state of a single photon in bulk optics by a special unbalanced beam splitter or by a balanced beam splitter accompanied by a state filtering. Also the all-fiber-based setup is discussed, where the information is encoded into spatial modes, i.e., the photon can propagate through two optical fibers. Each of the four implementations possesses some advantages and disadvantages that are discussed.
We report on two optical realizations of the $1\ensuremath{\rightarrow}2$ asymmetric phase-covariant cloning machines for polarization states of single photons. The experimental setups combine two-photon interference and tunable polarization filtering, which enables us to control the asymmetry of the cloners. The first scheme involves a special unbalanced bulk beam splitter exhibiting different splitting ratios for vertical and horizontal polarizations, respectively. The second implemented scheme consists of a balanced fiber coupler where photon bunching occurs, followed by a free-space part with polarization filters. With this latter approach we were able to demonstrate very high cloning fidelities which are above the universal cloning limit.
We study state-dependent quantum cloning that can outperform universal cloning (UC).This is possible by using some a priori information on a given quantum state to be cloned.Specifically, we propose a generalization and optical implementation of quantum optimal mirror phase-covariant cloning, which refers to optimal cloning of sets of qubits of known modulus of the expectation value of Pauli's Z operator.Our results can be applied to cloning of an arbitrary mirror-symmetric distribution of qubits on the Bloch sphere including in special cases UC and phase-covariant cloning.We show that the cloning is optimal by adapting our former optimality proof for axisymmetric cloning (Bartkiewicz and Miranowicz 2010 Phys.Rev. A 82 042330).Moreover, we propose an optical realization of the optimal mirror phase-covariant 1 → 2 cloning of a qubit, for which the mean probability of successful cloning varies from 1/6 to 1/3 depending on prior information on the set of qubits to be cloned.The qubits are represented by polarization states of photons generated by the type-I spontaneous parametric down-conversion.The scheme is based on the interference of two photons on an unbalanced polarization-dependent beam splitter with different splitting ratios for vertical and horizontal polarization components and the additional application of feedforward by means of Pockels cells.The experimental feasibility of the proposed setup is carefully studied including various kinds of imperfections and losses.Moreover, we briefly describe two possible cryptographic applications of the optimal mirror phase-covariant cloning corresponding to state discrimination (or estimation) and secure quantum teleportation.