Optimal economical telecloning of equatorial qubits
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Optimal economical telecloning of equatorial qubits Shi‑Jun Zhang1 · Wen‑Hai Zhang2 Received: 26 October 2019 / Accepted: 10 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We propose the optimal economical telecloning of equatorial qubits. The fidelity of each copy is optimal, and the telecloning can be realized with the success probability 100%. The efficiency of telecloning is much better than the protocol in the previous contributions (Wang and Yang in Phys Rev A 79:062315, 2009). Keywords Quantum cloning · Universal quantum cloning · Phase-covariant cloning · Real state cloning · Telecloning PACS Nos. 03.67.-a · 03.67.HK · 03.65.-w
1 Introduction In quantum mechanics, an arbitrary unknown quantum state cannot be cloned perfectly, which is called the no-cloning theorem [1]. This theorem provides theoretical support to the absolute security of quantum cryptography [2]. However, the no-cloning theorem does not exclude the attainability of imperfect copies by means of quantum operations allowed by quantum mechanics. Βužek and Hillery [3] first investigated the problem of how to obtain the best imperfect clones. Since then, quantum cloning becomes one of interesting topics in quantum communication [4]. In quantum cloning theory [4], the simplest quantum cloning machine is to clone ∑d−1 unknown pure quantum states in the form �𝜙⟩(in) = i=0 𝛼i ei𝛿i �i⟩ , where the real coef∑d−1 2 ficients 𝛼i satisfying i=0 𝛼i = 1 and the phase factors 𝛿i ∈ [0, 2𝜋) . If the amplitudes 𝛼i and the phase factors 𝛿i ∈ [0, 2𝜋) are completely unknown and the fidelity of each clone is independent to 𝛼i and 𝛿i ∈ [0, 2𝜋) , the cloning machine is the so-called universal quantum cloning (UQC) [3, 5–8]. If 𝛼i = d−1∕ 2 is known and 𝛿i ∈ [0, 2𝜋) is unknown, the cloner is defined as the phase-covariant cloning (PCC) [9–12]. If 𝛼i is unknown and 𝛿i = 0 is known, the cloner is the real state cloning (RSC) [13–15]. For * Wen‑Hai Zhang [email protected] 1
School of Electronic Engineering, Chaohu University, Chaohu 238000, Anhui, China
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School of Electronic Engineering, Huainan Normal University, Huainan 232038, Anhui, China
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the optimal cloning, the fidelity of each copy is maximal. Quantum cloning has many applications to quantum information science [16–19], and the PCC has an important application in quantum cryptography [2]. The UQC and PCC have been realized in experiment [20–23]. Quantum cloning can be realized not only locally, but also nonlocally. As the distribution of quantum information [24], Alice can transmit the imperfect clones to Bobs in distant cites by exploiting an entangled state shared among Alice and Bobs. This protocol is called telecloning [25], which is similar to quantum teleportation [26]. Telecloning has different types and was researched [27–29]. In Ref. [29], the authors proposed an economical telecloning of the phase-covariant state
� 1 � �𝜙⟩(in) = √ �0⟩ + ei𝛿 �1⟩ , 2
(1)
where 𝛿 ∈ [0, 2𝜋) a
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