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International Journal of Quantum Information

APPROXIMATE QUANTUM CLONING IN d DIMENSIONAL SPACE

Wen‐Hai Zhang, Liu Ye

Published May 30, 2007• 0 citations
Publisher

Abstract

We present a simpler method to deduce three types of approximate 1 → 2 quantum cloning machines in d dimensions, i.e. Wootters–Zurek, the universal, and phase-covariant quantum cloning machines. The explicit different transformations of these cloners are determined by different input states. Our method shows a relatively easy way to understand approximate quantum cloning.

DOI
10.1142/s0219749907002943

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We present a simpler method to deduce three types of approximate 1 → 2 quantum cloning machines in d dimensions, i.e. Wootters–Zurek, the universal, and phase-covariant quantum cloning machines. The explicit different transformations of these cloners are determined by different input states. Our method shows a relatively easy way to understand approximate quantum cloning.
Since the initial discovery of the Wootters-Zurek no-cloning theorem, a wide variety of quantum cloning machines have been proposed aiming at imperfect but optimal cloning of quantum states within its own context. Remarkably, most previous studies have employed the Bures fidelity or the Hilbert-Schmidt norm as the figure of merit to characterize the quality of the corresponding cloning scenarios. However, in many situations, what we truly care about is the relevant information about certain parameters encoded in quantum states. In this work, we investigate the multiple phase estimation problem in the framework of quantum cloning machines, from the perspective of quantum Fisher information matrix (QFIM). Focusing on the generalized ddd-dimensional equatorial states, we obtain the analytical formulas of QFIM for both universal quantum cloning machine (UQCM) and phase-covariant quantum cloning machine (PQCM), and prove that PQCM indeed performs better than UQCM in terms of QFIM. We highlight that our method can be generalized to arbitrary cloning schemes where the fidelity between the single-copy input and output states is input-state independent. Furthermore, the attainability of the quantum Cram\'er-Rao bound is also explicitly discussed.
Perfect cloning of an unknown quantum state is impossible. Approximate cloning, which is optimal in various senses, has been found in many cases. Paradigmatic examples are Wootters-Zurek cloning and universal cloning. These cloning machines aim at optimal cloning of the full quantum states. However, in practice, what is important and relevant may only involve partial information in quantum states, rather than quantum states themselves. For example, signals are often encoded as parameters in quantum states, whose information content is well synthesized by quantum Fisher information. This raises the basic issue of evaluating the information transferring capability (e.g., distributing quantum Fisher information) of quantum cloning. We assess and compare Wootters-Zurek cloning and universal cloning from this perspective and show that, on average, Wootters-Zurek cloning performs better than universal cloning for the phase (as well as amplitude) parameter, although they are incomparable individually, and universal cloning has many advantages over Wootters-Zurek cloning in other contexts. Physical insights and related issues are further discussed.
The possibility of cloning a ddd-dimensional quantum system without an ancilla is explored, extending on the economical phase-covariant cloning machine for qubits found in Phys. Rev. A 60, 2764 (1999). We prove the impossibility of constructing an economical version of the optimal universal $1\ensuremath{\rightarrow}2$ cloning machine in any dimension. We also show, using an ansatz on the generic form of cloning machines, that the ddd-dimensional $1\ensuremath{\rightarrow}2$ phase-covariant cloner, which optimally clones all balanced superpositions with arbitrary phases, can be realized economically only in dimension d=2d=2d=2. The used ansatz is supported by numerical evidence up to d=7d=7d=7. An economical phase-covariant cloner can nevertheless be constructed for d>2d>2d>2, albeit with a slightly lower fidelity than that of the optimal cloner requiring an ancilla. Finally, using again an ansatz on cloning machines, we show that an economical version of the $1\ensuremath{\rightarrow}2$ Fourier-covariant cloner, which optimally clones the computational basis and its Fourier transform, is also possible only in dimension d=2d=2d=2.
The study of quantum cryptography and quantum entanglement has traditionally been based on two-level quantum systems (qubits) and more recently on three-level systems (qutrits). We investigate several classes of state-dependent quantum cloners for four-level systems (quartits). These results apply to symmetric as well as asymmetric cloners, so that the balance between the fidelity of the two clones can also be analyzed. We extend Cerf's formalism for cloning states in order to derive cloning machines that remain invariant under certain unitary transformations. Our results show that a different cloner has to be used for two mutually unbiased bases which are related by a double Hadamard transformation, than for two mutually unbiased bases that are related by a Fourier transformation. This different cloner is obtained thanks to a redefinition of Bell states that respects the intrinsic symmetries of the Hadamard transformation.