Optimal asymmetric 1 → 4 quantum cloning in arbitrary dimension

We present an optimal asymmetric 1 → 4 quantum cloner. Our derivation generalizes the constructions of optimal asymmetric 1 → 2 and 1 → 3 quantum cloners in [Quantum Inf. Comput 5, 583 (2005)]. We explicitly prove the optimality of this cloner and give the maximum achievable fidelities. We also present the relation between the optimal quantum cloner with the multipartite entangled state which shows the singlet monogamy inequality in [Phys. Rev. Lett. 103, 050501 (2009)]. In quantum mechanics, the perfect copying of an unknown state is forbidden as a result of the linear superposition principle [1]. However, it is possible to construct approximate quantum cloning machines (QCM) yielding highfidelity copies. In 1996, Bužek and Hillery [2] constructed the optimal symmetric 1 → 2 universal QCM for qubits, soon afterwards its generalization to optimal symmetric m → n universal QCMs for both qubits and arbitrarydimensional systems were given [3–8]. All the above results were on symmetric cloning where all the copies have the same fidelity, however, from a more practical point especially in eavesdropping attack on certain kinds of QKD protocols [9–11], it is more interesting to consider optimal asymmetric cloning where the copies may have different fidelities. The family of optimal 1 → 2 asymmetric UQCMs for arbitrary-dimensional systems has been fully characterized [12–14]. However, for a long time, the proposed universal asymmetric cloning machines have only been conjectured to be optimal, and the proof of the optimality has been provided not long before [15–17]. In [17], Fiurášek et al. proved the optimality of universal asymmetric 1 → 2 cloning machines for qudits, they further extended the concept to quantum triplicators and constructed optimal asymmetric 1 → 3 cloning machine. In this paper, we will generalize their proof to an optimal asymmetric 1 → 4 cloning machine. In [18], Kay et al. related the optimal cloning with the singlet monogamy inequality in a multipartite system. In fact, the constraints on the shareability of maximal entanglement between multiple qudits is inherently linked with constraints on producing multiple copies of an unknown state. The singlet monogamy inequality in [18] was deduced from the best ansatz state for optimal asymmetric cloning. Here, we also a e-mail: [email protected] discuss the relations between their results and our optimal asymmetric 1 → 4 cloning machine. It should also be noted that, besides the theoretical developments on optimal quantum cloning, experimental implementation of various cloning machines has also been realized [19–23]. The organization of this paper is outlined as follows. Firstly, we formalize the general 1 → n asymmetric cloning problem in terms of the Jamio lkowski isomorphism. Then we briefly review the known optimal asymmetric 1 → 2 and 1 → 3 cloning machines. In the following, we make a through discussion on our optimal asymmetric 1 → 4 cloner and its relation with the singlet monogamy presented in reference [18]. Finally, we give a brief conclusion. In order to formalize the general 1 → n asymmetric cloning problem, we first introduce the Jamio lkowski isomorphism which gives a one-to-one correspondence between a completely positive (CP) map S from Hin to Hout and a positive operator S on Hin⊗ Hout. Let qudit 1 be the input and qudits 1, 2, . . . , n be the output copies. Then cloning operations correspond to completely positive maps S transforming qudit 1’s input state into output cloned states on qudits 1, 2, . . . , n. By introducing a reference qudit 0, we can construct Jamio lkowski isomorphism between completely positive maps S and positive semidefinite operators S ≥ 0, S = I ⊗ S(dΦ01), (1) here, d is the qudit’s dimension and Φ = |Φ+〉〈Φ+|, |Φ+〉 = 1 √ d ∑d−1 j=0 |j〉|j〉. Notice that the positive semidefinite operator S operates on qudits 0, 1, . . . , n. Since qudit 0 is a reference qudit, the completely positive map S should have no effect on it. At the same time, we assume that the completely positive map S is trace preserving. 2 The European Physical Journal D Therefore the positive semidefinite operator S should satisfy the constraints Tr1,...,n[S] = I0 and Tr[S] = d. In terms of the above isomorphism, we can formalize the fidelities of the general universal asymmetric cloning as follows. For an arbitrary input state |ψ〉, qudit k’s fidelity is Fk(ψ) = Tr[ψ 0 ⊗I1,...k̂...,n⊗ψk ·S]. Here, ψ = |ψ〉〈ψ| and k̂ represents the exclusion of k. Because we are considering universal cloning machines which clone equally well all states from the input Hilbert space, we should calculate the averaged fidelity Fk = ∫ ψ Fk(ψ)dψ = Tr[S · Lk]. Here Lk is given as