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arXiv:quant-ph/9607018 · Physical Review A

Quantum copying: Beyond the no-cloning theorem

Vladimír Bužek, Mark S. Hillery

Published Sep 1, 1996• 1,124 citations
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Abstract

We analyze the possibility of copying (that is, cloning) arbitrary states of a quantum-mechanical spin-1/2 system. We show that there exists a ``universal quantum-copying machine'' (i.e., transformation) which approximately copies quantum-mechanical states such that the quality of its output does not depend on the input. We also examine a machine which combines a unitary transformation and a selective measurement to produce good copies of states in the neighborhood of a particular state. We discuss the problem of measurement of the output states. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.1844

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Bužek and M. Hillery analyze the possibility of (e) copying (that is, cloning) arbitrary states of (e) a quantum-mechanical spin-1/2 system. Authors show that there exists (eb) universal quantum-copying machine‘‘ (i.e., transformation) which approximately copies quantum-mechanical states such that (e) the quality of its output does not (e) depend on the input. Authors also examine a machine which combines a unitary transformation and (e&eb) a selective measurement to produce good copies of (e) states in the neighborhood of a particular state. Authors discuss the problem of (e) measurement of the output states. © 1996 The American Physical Society Received 5 February 1996 DOI:http://dx.doi.org/10.1103/PhysRevA.54.1844 Quantum copying: Beyond the no-cloning theorem V. Bužek and M. Hillery Phys. Rev. A 54, 1844 – Published 1 September 1996 STOP Authors derive a necessary and sufficient condition for two pure states, each entangled in two remote systems, to be clonable by the sequential access to the two systems. The result shows (eb) to what extent... The full paper: http://www.iiste.org/PDFshare/APTA-PAGENO-496948-501458.pdf

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We analyze the possibility of copying (that is, cloning) arbitrary states of a quantum-mechanical spin-1/2 system. We show that there exists a ``universal quantum-copying machine'' (i.e., transformation) which approximately copies quantum-mechanical states such that the quality of its output does not depend on the input. We also examine a machine which combines a unitary transformation and a selective measurement to produce good copies of states in the neighborhood of a particular state. We discuss the problem of measurement of the output states. \textcopyright{} 1996 The American Physical Society.
Bužek and M. Hillery analyze the possibility of (e) copying (that is, cloning) arbitrary states of (e) a quantum-mechanical spin-1/2 system. Authors show that there exists (eb) universal quantum-copying machine‘‘ (i.e., transformation) which approximately copies quantum-mechanical states such that (e) the quality of its output does not (e) depend on the input. Authors also examine a machine which combines a unitary transformation and (e&eb) a selective measurement to produce good copies of (e) states in the neighborhood of a particular state. Authors discuss the problem of (e) measurement of the output states. © 1996 The American Physical Society Received 5 February 1996 DOI:http://dx.doi.org/10.1103/PhysRevA.54.1844 Quantum copying: Beyond the no-cloning theorem V. Bužek and M. Hillery Phys. Rev. A 54, 1844 – Published 1 September 1996 STOP Authors derive a necessary and sufficient condition for two pure states, each entangled in two remote systems, to be clonable by the sequential access to the two systems. The result shows (eb) to what extent... The full paper: http://www.iiste.org/PDFshare/APTA-PAGENO-496948-501458.pdf
We analyze how information encoded in quantum systems can be optimally processed. In particular, we investigate copying (cloning) of quantum information (represented as states of 2-level quantum systems--qubits). We present unitary transformations which describe the optimal universal cloning of a qubit. Universality of the transformation guarantees that the fidelity of the cloning does not depend on the input state of the qubit, i.e. all states are cloned equally well. We present network for the optimal universal quantum cloning `machine' (transformation) which produces N + 1 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 copes, N. We also present the `machine' which universally and optimally clones states of quantum objects in arbitrary- dimensional Hilbert spaces. In particular, we discuss universal cloning of quantum registers. In addition to cloning of qubits we analyzed another universal operation-- the Universal NOT. We present the optimal transformation and the corresponding logical network which optimally complements an arbitrary input state of a qubit. We show that the fidelity of the performance of the Universal NOT operation increases as a function of the number of input qubits prepared in the same state.
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 quantum cloning machines that transform NNN identical qubits into M>NM>NM>N identical copies and we prove that the fidelity (quality) of these copies is optimal. The connection between cloning and measurement is discussed in detail. When the number of clones MMM tends towards infinity, the fidelity of each clone tends towards the optimal fidelity that can be obtained by a measurement on the input qubits. More generally, quantum cloning machines are universal devices to translate quantum information into classical information.
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.
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.