Designing optimal quantum cloning machine for qubit system

TL;DR

This paper develops a systematic method for designing optimal quantum cloning machines for qubits using Bloch-sphere representation and Fiurášek's optimality criterion, unifying various known cloning protocols.

Contribution

It introduces a general scheme for constructing unitary transformations for optimal quantum cloning, encompassing multiple existing cloning strategies and criteria.

Findings

Recovered known optimal fidelities for various input ensembles.

Provided a unified framework for different cloning protocols.

Presented a method to construct unitary transformations for optimal cloning.

Abstract

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 Fiur\'{a}\u{s}ek'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.