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Physical Review A

Optimal quantum cloning based on the maximin principle by usinga prioriinformation

Peng Kang, Hong‐Yi Dai, Jiahua Wei, Ming Zhang

Published Oct 5, 2016• 9 citations
Publisher

Abstract

We propose an optimal $1\ensuremath{\rightarrow}2$ quantum cloning method based on the maximin principle by making full use of a priori information of amplitude and phase about the general cloned qubit input set, which is a simply connected region enclosed by a ``longitude-latitude grid'' on the Bloch sphere. Theoretically, the fidelity of the optimal quantum cloning machine derived from this method is the largest in terms of the maximin principle compared with that of any other machine. The problem solving is an optimization process that involves six unknown complex variables, six vectors in an uncertain-dimensional complex vector space, and four equality constraints. Moreover, by restricting the structure of the quantum cloning machine, the optimization problem is simplified as a three-real-parameter suboptimization problem with only one equality constraint. We obtain the explicit formula for a suboptimal quantum cloning machine. Additionally, the fidelity of our suboptimal quantum cloning machine is higher than or at least equal to that of universal quantum cloning machines and phase-covariant quantum cloning machines. It is also underlined that the suboptimal cloning machine outperforms the ``belt quantum cloning machine'' for some cases.

DOI
10.1103/physreva.94.042304

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We propose an optimal 1\ensuremath→21\ensuremath{\rightarrow}21\ensuremath→2 quantum cloning method based on the maximin principle by making full use of a priori information of amplitude and phase about the general cloned qubit input set, which is a simply connected region enclosed by a longitude-latitude grid'' on the Bloch sphere. Theoretically, the fidelity of the optimal quantum cloning machine derived from this method is the largest in terms of the maximin principle compared with that of any other machine. The problem solving is an optimization process that involves six unknown complex variables, six vectors in an uncertain-dimensional complex vector space, and four equality constraints. Moreover, by restricting the structure of the quantum cloning machine, the optimization problem is simplified as a three-real-parameter suboptimization problem with only one equality constraint. We obtain the explicit formula for a suboptimal quantum cloning machine. Additionally, the fidelity of our suboptimal quantum cloning machine is higher than or at least equal to that of universal quantum cloning machines and phase-covariant quantum cloning machines. It is also underlined that the suboptimal cloning machine outperforms the belt quantum cloning machine'' for some cases.
We find an optimal quantum cloning machine, which clones qubits of arbitrary symmetrical distribution around the Bloch vector with the highest fidelity. The process is referred to as phase-independent cloning in contrast to the standard phase-covariant cloning for which an input qubit state is a priori better known. We assume that the information about the input state is encoded in an arbitrary axisymmetric distribution (phase function) on the Bloch sphere of the cloned qubits. We find analytical expressions describing the optimal cloning transformation and fidelity of the clones. As an illustration, we analyze cloning of qubit state described by the von Mises--Fisher and Brosseau distributions. Moreover, we show that the optimal phase-independent cloning machine can be implemented by modifying the mirror phase-covariant cloning machine for which quantum circuits are known.
Following the work of Niu and Griffiths, in \emph{Phys.Rev.A 58, 4377(1998)}, we shall investigate the problem, how to design the optimal quantum cloning machines (QCMs) for qubit system, with the help of Bloch-sphere representation. In stead of the quality factor there, the Fiuráusek's optimal condition, where the optimal cloning machine should maximize a convex mixture of the average fidelity, is used as the optimality criterion in present protocol. Almost all of the known optimal QCMs in previous works, the cloning for states with fixed polar angle, the phase-covariant cloning, the universal QCMs, the cloning for two arbitrary pure states, and the mirror phase-covariant cloning, should be discussed in a systematic way. The known results, the optimal fidelities for various input ensembles according to different optimality criteria, are recovered here. Our present scheme also offers a general way of constructing the unitary transformation to realize the optimal cloning.
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 propose a quantum cloning machine, which clones a qubit into two clones assuming known modulus of expectation value of Pauli ${\ensuremath{\sigma}}_{z}$ matrix. The process is referred to as the mirror phase-covariant cloning for which the input state is a priori less known than that for the standard phase-covariant cloning. Analytical expressions describing the cloning transformation and fidelity of the clones are found. Extremal equations for the optimal cloning are derived and analytically solved by generalizing a method of Fiur\'a\ifmmode \check{s}\else \v{s}\fi{}ek [Phys. Rev. A 64, 062310 (2001)]. Quantum circuits implementing the optimal cloning transformation and their physical realization in a quantum-dot system are described.
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.
After the appearance of the no-cloning theorem, approximate quantum cloning machines (QCMs) has become a well-studied subject in quantum information theory. Among several measures to quantify the performance of a QCM, single-qudit fidelity and global fidelity have been most widely used. In this paper we compute the optimal global fidelity for phase-covariant cloning machines via semidefinite programming optimization, thereby completing a remaining gap in the previous results on QCMs. We also consider optimal simulations of the cloning and transpose cloning map, both by a direct optimization and by a composition of component-wise optimal QCMs. For the cloning map the composition method is suboptimal whereas for the transpose cloning map the method is asymptotically optimal.