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arXiv:quant-ph/0412041 · Physical Review A

Realization of the optimal phase-covariant quantum cloning machine

Fabio Sciarrino, F. De Martini

Published Dec 9, 2005• 48 citations
arXivHTMLPDF

Abstract

In several quantum information (QI) phenomena of large technological importance the information is carried by the phase of the quantum superposition states, or qubits. The phase-covariant cloning machine (PQCM) addresses precisely the problem of optimally copying these qubits with the largest attainable ``fidelity.'' We present a general scheme which realizes the $1\ensuremath{\rightarrow}3$ phase covariant cloning process by a combination of three different QI processes: the universal cloning, the NOT gate, and the projection over the symmetric subspace of the output qubits. The experimental implementation of a PQCM for polarization encoded qubits, the first ever realized with photons, is reported.

DOI
10.1103/physreva.72.062313

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In several quantum information (QI) phenomena of large technological importance the information is carried by the phase of the quantum superposition states, or qubits. The phase-covariant cloning machine (PQCM) addresses precisely the problem of optimally copying these qubits with the largest attainable ``fidelity.'' We present a general scheme which realizes the 1\ensuremath→31\ensuremath{\rightarrow}31\ensuremath→3 phase covariant cloning process by a combination of three different QI processes: the universal cloning, the NOT gate, and the projection over the symmetric subspace of the output qubits. The experimental implementation of a PQCM for polarization encoded qubits, the first ever realized with photons, is reported.
The optimal phase-covariant quantum cloning machine (PQCM) broadcasts the information associated to an input qubit into a multiqubit system, exploiting a partial a priori knowledge of the input state. This additional a priori information leads to a higher fidelity than for the universal cloning. The present article first analyzes different innovative schemes to implement the $1\ensuremath{\rightarrow}3$ PQCM. The method is then generalized to any $1\ensuremath{\rightarrow}M$ machine for an odd value of MMM by a theoretical approach based on the general angular momentum formalism. Finally different experimental schemes based either on linear or nonlinear methods and valid for single photon polarization encoded qubits are discussed.
We experimentally realize the $1\ensuremath{\rightarrow}2$ optimal universal quantum cloning machine for arbitrary qubits and the ancilla-free $1\ensuremath{\rightarrow}3$ optimal phase-covariant quantum cloning machine for $x\text{\ensuremath{-}}y$ equatorial qubits, which were constructed by Zou et al. [Phys. Lett. A 366, 36 (2007)]. The qubits are represented by the polarization states of photons prepared from the process of spontaneous parametric down-conversion. The copies of each machine are shown to have almost equal fidelities conditioned on fourfold coincidences.
Since the initial discovery of the Wootters-Zurek no-cloning theorem, a wide variety of quantum cloning machines have been proposed aiming at imperfect but optimal cloning of quantum states within its own context. Remarkably, most previous studies have employed the Bures fidelity or the Hilbert-Schmidt norm as the figure of merit to characterize the quality of the corresponding cloning scenarios. However, in many situations, what we truly care about is the relevant information about certain parameters encoded in quantum states. In this work, we investigate the multiple phase estimation problem in the framework of quantum cloning machines, from the perspective of quantum Fisher information matrix (QFIM). Focusing on the generalized ddd-dimensional equatorial states, we obtain the analytical formulas of QFIM for both universal quantum cloning machine (UQCM) and phase-covariant quantum cloning machine (PQCM), and prove that PQCM indeed performs better than UQCM in terms of QFIM. We highlight that our method can be generalized to arbitrary cloning schemes where the fidelity between the single-copy input and output states is input-state independent. Furthermore, the attainability of the quantum Cram\'er-Rao bound is also explicitly discussed.
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.
While exact cloning of an unknown quantum state is prohibited by the linearity of quantum mechanics, approximate cloning is possible and has been used, e.g., to derive limits on the security of quantum communication protocols. In the case of asymmetric cloning, the information from the input state is distributed asymmetrically between the different output states. Here, we consider asymmetric phase-covariant cloning, where the goal is to optimally transfer the phase information from a single input qubit to different output qubits. We construct an optimal quantum cloning machine for two qubits that does not require ancilla qubits and implement it on an NMR quantum information processor.