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International Journal of Quantum Information

"PARTIAL" QUANTUM CLONING AND QUANTUM CLONING OF MIXED STATES

Alexander Ya. Kazakov

Published Apr 1, 2010• 3 citations
Publisher

Abstract

We discuss "partial" quantum cloning of pure two-partite states, where only the "part" of the initial state related to one qubit is copied. The same approach gives the possibility of designing a quantum copying machine for mixed qubit states.
DOI
10.1142/s0219749910006186

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Quantum cloning of two identical mixed qubits ρ⊗ρ\rho \otimes \rhoρ⊗ρ is studied. We propose the quantum cloning transformations not only for the triplet (symmetric) states but also for the singlet (antisymmetric) state. We can copy these two identical mixed qubits to MMM (M≥2M\ge 2M≥2) copies. This quantum cloning machine is optimal in the sense that the shrinking factor between the input and the output single qubit achieves the upper bound. The result shows that we can copy two identical mixed qubits with the same quality as that of two identical pure states.
It is a well-known fact that the no-cloning theorem forbids the creation of identical copies of an arbitrary unknown quantum state.In other words,there does not exist a quantum cloning machine that can clone all quantum states.However,it is possible to clone given quantum states under certain conditions,for instance,k distinct pure states|ψ1,|ψ2,...,|ψk can be cloned simultaneously if and only if they are orthogonal.This paper discusses the existence and construction of simultaneous cloning machines for mixed states.It is proved that k distinct mixed statesρ1,ρ2,...,ρkof the n-dimensional quantum system Cncan be cloned simultaneously,that is,there exists a quantum channel Φ on MnMn and a state Σ in Mn,such that Φ(ρiΣ)=ρiρi for all i,if and only ifρiρj=0(i j).Also,the constructing procedure of the desired simultaneous cloning machine is given.
We present a unified universal quantum cloning machine, which combines several different existing universal cloning machines together, including the asymmetric case. In this unified framework, the identical pure states are projected equally into each copy initially constituted by input and one half of the maximally entangled states. We show explicitly that the output states of those universal cloning machines are the same. One importance of this unified cloning machine is that the cloning procession is always the symmetric projection, which reduces dramatically the difficulties for implementation. Also, it is found that this unified cloning machine can be directly modified to the general asymmetric case. Besides the global fidelity and the single-copy fidelity, we also present all possible arbitrary-copy fidelities.