Realization of a General Multi-step Quantum Cloning Machine

A general multi-step N 7−→ M probabilistic optimal universal cloning protocol is presented together with the experimental realization of the (1 → 3) and (2 → 3) machines. 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 applications as a novel, multi-qubit symmetrizator device to be used in modern quantum information networks. Typeset using REVTEX 1 A most relevant limitation in quantum information processing is the impossibility of perfectly cloning (copying) any unknown qubit |φ〉 [1]. Even if this process is unrealizable in its exact form, it can be approximated optimally by the so-called universal optimal quantum cloning machine (UOQCM), one which exhibits the minimum possible noise for any possible input state. From a theoretical perspective, two different kinds of universal cloning machines have been developed so far: a deterministic N → M UOQCM based on a unitary operator acting on N input qubits and 2(M − N) ancilla qubits [2,3] and a probabilistic UOQCM based on a symmetrization procedure involving a projective operator acting on N inputs and (M −N) blank ancilla qubits [4]. In the last years several experimental realizations of the UOQCM for polarization (π−) encoded photon qubits have been reported. The deterministic UOQCM has been realized by associating the cloning effect with QED stimulated emission [5] while the probabilistic machine has been realized adopting a linear symmetrization protocol [6]. Thus far, only the simplest 1 → 2 cloning processes, i.e. for N = 1 and M = 2, were realized by both schemes. In particular, the probabilistic process was achieved by exploiting the bosonic character of the photons within a linear Hong-Ou-Mandel interferometer scheme [7]. The present work presents the first generalization of the universal optimal cloning process by the realization of a very general linear procedure to extend the probabilistic protocol to any value of N and M according to a suggestion by Werner. [4]. The validity of this theoretical scheme is supported by the here reported experimental implementations of the 1 → 3 and 2 → 3 probabilistic processes for π−encoded photon qubits (π−qubits). Let us outline first the N → M probabilistic cloning theory. Consider N identically prepared unknown qubits in the state ρi = |φ〉 〈φ| as input of the cloning machine while (M − N) blank qubits, i.e. all in the state ρA = I 2 , are used as an auxiliary resource. To generate M output clones the machine performs the symmetrization of the output state by applying the projective operator, Π (M) + over the symmetric subspace of M qubits: |φ〉 〈φ| N→M −→ 1 pN→M [Π (M) + (|φ〉 〈φ| ⊗N ⊗ I 2 ⊗(M−N) )Π (M) + ] (1)