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arXiv:quant-ph/9712019 · Physical Review Letters

Optimal Universal Quantum Cloning and State Estimation

Dagmar Bruß, ARTUR K. EKERT, Chiara Macchiavello

Published Sep 21, 1998• 283 citations
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Abstract

We derive a tight upper bound for the fidelity of a universal $N\ensuremath{\rightarrow}M$ qubit cloner, valid for any $M\ensuremath{\ge}N$, where the output of the cloner is required to be supported on the symmetric subspace. Our proof is based on the concatenation of two cloners and the connection between quantum cloning and quantum state estimation. We generalize the operation of a quantum cloner to mixed and/or entangled input qubits described by a density matrix supported on the symmetric subspace of the constituent qubits. We also extend the validity of optimal state estimation methods to inputs of this kind.

DOI
10.1103/physrevlett.81.2598

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We derive a tight upper bound for the fidelity of a universal N\ensuremath→MN\ensuremath{\rightarrow}MN\ensuremath→M qubit cloner, valid for any M\ensuremath≥NM\ensuremath{\ge}NM\ensuremath≥N, where the output of the cloner is required to be supported on the symmetric subspace. Our proof is based on the concatenation of two cloners and the connection between quantum cloning and quantum state estimation. We generalize the operation of a quantum cloner to mixed and/or entangled input qubits described by a density matrix supported on the symmetric subspace of the constituent qubits. We also extend the validity of optimal state estimation methods to inputs of this kind.
We pose the question whether the asymptotic equivalence between quantum cloning and quantum state estimation, valid at the single-clone level, still holds when all clones are examined globally. We conjecture that the answer is affirmative and present a large amount of evidence supporting our conjecture, developing techniques to derive optimal asymptotic cloners and proving their equivalence with estimation in virtually all scenarios considered in the literature. Our analysis covers the case of arbitrary finite sets of states, arbitrary families of coherent states, arbitrary phase- and multiphase-covariant sets of states, and two-qubit maximally entangled states. In all these examples we observe that the optimal asymptotic cloners enjoy a universality property, consisting in the fact that scaling of their fidelity does not depend on the specific details of the input states, but only on the number of free parameters needed to specify them.
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.
We investigate the asymptotic relationship between quantum cloning and quantum estimation from the global point of view where all the copies produced by the cloner are considered jointly. For an N-to-M cloner, we consider the overall fidelity between the state of the M output systems and the state of M ideal copies, and we ask whether the optimal fidelity is attained by a measure and-prepare protocol in the limit M -> \infty. In order to gain intuition into the general problem, we analyze two concrete examples: i) cloning qubit states on the equator of the Bloch sphere and ii) cloning two-qubit maximally entangled states. In the first case, we show that the optimal measure-and-prepare fidelity converges to the fidelity of the optimal cloner in the limit M -> \infty. In the second case, we restrict our attention to economical covariant cloners, and again, we exhibit a measure-and-prepare protocol that achieves asymptotically the optimal fidelity. Quite counterintuitively, in both cases the optimal states that have to be prepared in order to maximize the overall fidelity are not product states corresponding to M identical copies, but instead suitable M-partite entangled states.
We show that no universal quantum cloning machine exists that can broadcast an arbitrary mixed qubit with a constant fidelity. Based on this result, we investigate the dependent quantum cloner in the sense that some parameter of the input qubit ${\ensuremath{\rho}}_{s}(\ensuremath{\theta},\ensuremath{\omega},\ensuremath{\lambda})$ is regarded as constant in the fidelity. For the case of constant $\ensuremath{\omega}$, we establish the $1\ensuremath{\rightarrow}2$ optimal symmetric dependent cloner with a fidelity 1∕21∕21∕2. It is also shown that the $1\ensuremath{\rightarrow}M$ optimal quantum cloning machine for pure qubits is also optimal for mixed qubits, when $\ensuremath{\lambda}$ is the unique parameter in the fidelity. For general $N\ensuremath{\rightarrow}M$ broadcasting of mixed qubits, the situation is very different.