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arXiv:quant-ph/0407176 · Physical Review A

Realization of universal optimal quantum machines by projective operators and stochastic maps

Fabio Sciarrino, Carlo Sias, Marco Ricci, F. De Martini

Published Nov 1, 2004• 35 citations
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Abstract

Optimal quantum machines can be implemented by linear projective operations. In the present work a general qubit symmetrization theory is presented by investigating the close links to the qubit purification process and to the programmable teleportation of any generic optimal antiunitary map. In addition, the contextual realization of the $N\ensuremath{\rightarrow}M$ cloning map and of the teleportation of the $N\ensuremath{\rightarrow}(M\ensuremath{-}N)$ universal-NOT (UNOT) gate is analyzed by a very general angular momentum theory. An extended set of experimental realizations by state symmetrization linear optical procedures is reported. These include the $1\ensuremath{\rightarrow}2$ cloning process, the UNOT gate and the quantum tomographic characterization of the optimal partial transpose map of polarization encoded qubits.

DOI
10.1103/physreva.70.052305

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Optimal quantum machines can be implemented by linear projective operations. In the present work a general qubit symmetrization theory is presented by investigating the close links to the qubit purification process and to the programmable teleportation of any generic optimal antiunitary map. In addition, the contextual realization of the N\ensuremath→MN\ensuremath{\rightarrow}MN\ensuremath→M cloning map and of the teleportation of the N\ensuremath→(M\ensuremath−N)N\ensuremath{\rightarrow}(M\ensuremath{-}N)N\ensuremath→(M\ensuremath−N) universal-NOT (UNOT) gate is analyzed by a very general angular momentum theory. An extended set of experimental realizations by state symmetrization linear optical procedures is reported. These include the 1\ensuremath→21\ensuremath{\rightarrow}21\ensuremath→2 cloning process, the UNOT gate and the quantum tomographic characterization of the optimal partial transpose map of polarization encoded qubits.
By a significant modification of the standard protocol of quantum state teleportation, two processes "forbidden" by quantum mechanics in their exact form, the universal NOT gate and the universal optimal quantum cloning machine, have been implemented contextually and optimally by a fully linear method. In particular, the first experimental demonstration of the tele-UNOT gate, a novel quantum information protocol, has been reported. The experimental results are found in full agreement with theory.
Abstract Since manipulations of qubits are constrained by the quantum mechanical rules, several classical information tasks can not be perfectly extended to the quantum world. The more relevant limitations in quantum information processing are the impossibility to perfectly clone any unknown qubit and to map it in its orthogonal state. Even if these two processes are unrealizable in their exact forms, they can be optimally approximated by the so‐called universal quantum machines: the optimal quantum cloning machine (UOQCM) and the universal‐NOT (U‐NOT) gate. Investigation of these optimal transformations is important since it reveals bounds on optimal manipulations of information with quantum systems. The 1−>1 U‐NOT gate and the 1−>2 UOQCM have been experimentally demonstrated by adopting the process of stimulated emission into an optical parametric amplifier and by slightly modifying the quantum state teleportation protocol.
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.
In this paper, in a simple unifying matrix framework, I will present general classification of all possible tomography methods, teleportation schemes, and optimal quantum cloning maps. We will see how every tomographic method or teleportation scheme corresponds to a choice of operator spanning sets, and how this framework also leads to methods for engineering new Bell measurements. On the other hand, the classification of all possible covariant cloning maps (that are optimal for a given criterion) includes all known types of cloning, and leads to methods for engineering new cloning machines, which can be physically realized through unitary transformation with ancilla, and/or via probabilistic quantum operations. Fidelity criteria for POVM’s can be exploited to achieve joint POVM’s via cloning. I will give concrete physical realizations in the paper.
A general multistep linear state symmetrization device for photonic qubits is presented together with the experimental realizations of the $1\ensuremath{\rightarrow}3$ and $2\ensuremath{\rightarrow}3$ universal optimal quantum cloning machines and of a 3-qubit purification procedure. Since the present method exploits the bosonic nature of the photons, it can be applied to any particle obeying to the Bose statistics. On a technological perspective, the present protocol is expected to find relevant applications as a multiqubit symmetrization device to be used in modern quantum-information networks.
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.