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Journal of Atomic and Molecular Physics

Optimality of the Asymmetric Economical phase-covariant Quantum Cloning

Jie-Lin Dai, Wen‐Hai Zhang

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

Quantum cloning can be reviewed as the optimal individual eavesdropping on quantum cryptography. Specifically, the optimal phase-covariant cloning is regarded as the optimal individual attack on the BB84 protocol. In this paper, by exploring Cerf s formalism, we prove the optimality of the asymmetric economical 1→2 phase-covariant quantum cloning in 2 dimensions working without ancilla. In proof, we derive another independent unitary transformation of this cloner. Our result covers previous contributions [Durt T, et al. Phys. Rev. A, 2004, 69:062316].

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AIP Advances•Dec 1, 2024•0 citations
Quantum cloning can be reviewed as the optimal individual eavesdropping on quantum cryptography. Specifically, the optimal phase-covariant cloning is regarded as the optimal individual attack on the BB84 protocol. In this paper, by exploring Cerf s formalism, we prove the optimality of the asymmetric economical 1→2 phase-covariant quantum cloning in 2 dimensions working without ancilla. In proof, we derive another independent unitary transformation of this cloner. Our result covers previous contributions [Durt T, et al. Phys. Rev. A, 2004, 69:062316].
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.
Though the BB84 protocol has provable security over a noiseless quantum channel, the security is not proven over current noisy technology. The level of tolerable error on such systems is still unclear, as is how much information about a raw key may be obtained by an eavesdropper. We develop a reproducible test to determine the security–or lack thereof–of the protocol in practice. This enables us to obtain an experimental estimate of the information that can be obtained using asymmetric phase-covariant cloning to eavesdrop on the BB84 protocol.
While exact cloning of an unknown quantum state is prohibited by the linearity of quantum mechanics, approximate cloning is possible and has been used, e.g., to derive limits on the security of quantum communication protocols. In the case of asymmetric cloning, the information from the input state is distributed asymmetrically between the different output states. Here, we consider asymmetric phase-covariant cloning, where the goal is to optimally transfer the phase information from a single input qubit to different output qubits. We construct an optimal quantum cloning machine for two qubits that does not require ancilla qubits and implement it on an NMR quantum information processor.
The security of quantum cryptography is guaranteed by the no-cloning theorem, which implies that an eavesdropper copying transmitted qubits in unknown states causes their disturbance. Nevertheless, in real cryptographic systems some level of disturbance has to be allowed to cover, e.g., transmission losses. An eavesdropper can attack such systems by replacing a noisy channel by a better one and by performing approximate cloning of transmitted qubits which disturb them but below the noise level assumed by legitimate users. We experimentally demonstrate such symmetric individual eavesdropping on the quantum key distribution protocols of Bennett and Brassard (BB84) and the trine-state spherical code of Renes (R04) with two-level probes prepared using a recently developed photonic multifunctional quantum cloner [Lemr et al., Phys. Rev. A 85, 050307(R) (2012)]. We demonstrated that our optimal cloning device with high-success rate makes the eavesdropping possible by hiding it in usual transmission losses. We believe that this experiment can stimulate the quest for other operational applications of quantum cloning.