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arXiv:0804.0129 · Physical Review Letters

Optimal Cloning of Unitary Transformation

Giulio Chiribella, Giacomo Mauro D’Ariano, Paolo Perinotti

Published Oct 30, 2008• 69 citations
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Abstract

After proving a general no-cloning theorem for black boxes, we derive the optimal universal cloning of unitary transformations, from one to two copies. The optimal cloner is realized by quantum channels with memory, and greatly outperforms the optimal measure-and-reprepare cloning strategy. Applications are outlined, including two-way quantum cryptographic protocols.

DOI
10.1103/physrevlett.101.180504

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After proving a general no-cloning theorem for black boxes, we derive the optimal universal cloning of unitary transformations, from one to two copies. The optimal cloner is realized by quantum channels with memory, and greatly outperforms the optimal measure-and-reprepare cloning strategy. Applications are outlined, including two-way quantum cryptographic protocols.
We establish the best possible approximation to a perfect quantum cloning machine that produces two clones out of a single input. We analyze both universal and state-dependent cloners. The maximal fidelity of cloning is shown to be 5/6 for universal cloners. It can be achieved either by a special unitary evolution or by a teleportation scheme. We construct the optimal state-dependent cloners operating on any prescribed two nonorthogonal states and discuss their fidelities and the use of auxiliary physical resources in the process of cloning. The optimal universal cloners permit us to derive an upper bound on the quantum capacity of the depolarizing quantum channel.
The no-cloning theorem is one of the fundamental concepts of quantum information theory. It tells us that no general quantum state can be perfectly replicated. In this thesis we introduce the notion of imperfect cloning, and define the properties of the universal cloning machine. Furthermore, we construct an ansatz of how our universal cloning machine should perform as to produce two imperfect clones from one input qubit. We find an optimal fidelity of 5/6 for our universal cloning machine. We then reevaluate our ansatz and construct a class of unitary transformations such that an optimal fidelity is always achieved. Lastly, we present an overview of some applications of imperfect quantum cloning in the field of quantum information.
Cloning machines, that is, transformations that achieve the best approximate copying of a quantum state compatible with the no-cloning theorem, have been a fundamental research topic over the last five years. This study is of particular significance given the close connection between quantum cloning and quantum cryptography: using an optimal cloner generally makes it possible to obtain a tight bound on the best individual eavesdropping strategy in a quantum cryptosystem. In this Chapter, the issue of cloning a continuous-variable quantum system will be analyzed, and a Gaussian cloning transformation will be introduced. This cloner, which copies equally well any two canonically conjugate continuous variables, duplicates all coherent states with a same fidelity (F=2/3). The optical implementation of this cloner and its extension to N-to-M cloners will also be discussed. Finally, the use of this cloner for the security assessment of a quantum key distribution scheme relying on continuous (Gaussian) key carriers will be sketched.
We review our recent work on the universal (i.e. input state independent) optimal quantum copying (cloning) of qubits. We present unitary transformations which describe the optimal cloning of a qubit and we present the corresponding quantum logical network. We also present network for an optimal quantum copying ``machine'' (transformation) which produces N+1 identical copies from the original qubit. Here again the quality (fidelity) of the copies does not depend on the state of the original and is only a function of the number of copies, N. In addition, we present the machine which universaly and optimally clones states of quantum objects in arbitrary-dimensional Hilbert spaces. In particular, we discuss universal cloning of quantum registers.
We present the universal cloning transformation of states in arbitrary-dimensional Hilbert spaces. This unitary transformation attains the optimal fidelity of cloning as specified by Werner [Phys. Rev. A 58, 1827 (1998)]. With this cloning transformation, pure as well as impure states can be optimally copied, and the quality of the copies does not depend on the state being copied. We discuss the properties of quantum clones. In particular, we show that in the limit of high dimension the fidelity of clones does not converge to zero but attains the limit 1/21/21/2. We also show that our cloning transformation is most suitable for cloning of entanglement.