In a previous paper [Phys. Rev. A69 (2004) 062316], two of us (JD and TD) derived a necessary and sufficient condition for the reducibility of 3 qubit one-to-two cloners to 2 qudit one-to-two cloners. When this criterion is fulfilled, economical cloning is possible. We show in the present paper how to generalize the assumptions under which this condition is valid. We also extend this result to qudit one-to-two cloners, with d finite but arbitrary.
In a previous paper [Phys. Rev. A69 (2004) 062316], two of us (JD and TD) derived a necessary and sufficient condition for the reducibility of 3 qubit one-to-two cloners to 2 qudit one-to-two cloners. When this criterion is fulfilled, economical cloning is possible. We show in the present paper how to generalize the assumptions under which this condition is valid. We also extend this result to qudit one-to-two cloners, with d finite but arbitrary.
In this paper we establish a deep connection between the 3-qubit one-to-two phase-covariant quantum cloning network of Fuchs et al. [Phys. Rev. A 56, 1163 (1997)], and its low-cost two-qubit counterpart due to Niu and Griffiths [Phys. Rev. A 60, 2764 (1999)]. A general, necessary, and sufficient criterion is derived in order to characterize the reducibility of 3-qubit one-to-two cloners to two-qubit one-to-two cloners. When this criterion is fulfilled, low-cost cloning is possible. We show that the optimal isotropic or universal 3-qubit one-to-two cloning machine is not reducible to a 2-qubit one-to-two cloner.
In this paper we establish a deep connection between the 3 qubit one-to-two phase-covariant quantum cloning network of Fuchs et al. [C. Fuchs, N. Gisin, R.B. Griffiths, C.S. Niu, and A. Peres, Phys. Rev. A 56 no 4, 1163 (1997)], and its economic 2 qubit counterpart due to Niu and Griffiths [Phys.Rev. A 60 no 4, 2764 (1999)]. A general, necessary and sufficient criterion is derived in order to characterize the reducibility of 3 qubit cloners to 2 qubit cloners. When this criterion is fulfilled, economic cloning is possible. We show that the optimal isotropic or universal 3 qubit cloning machine is not reducible to a 2 qubit cloner.
While the no-cloning theorem, which forbids the perfect copying of quantum states, is well-known as one of the defining features of quantum mechanics, the question of how well the theory allows a state to be cloned is yet to be completely solved. In this paper, rigorous solutions to the problem of M to N asymmetric cloning of qudits are obtained in a number of interesting cases. The central result is the solution to the 1 to N universal asymmetric qudit cloning problem for which the exact trade-off in the fidelities of the clones for every N and d is derived. Analogous results are proven for qubits when M=N-1. We also consider state-dependent 1 to N qubit cloning, providing a general parametrization in terms of a Heisenberg star Hamiltonian. In all instances, we determine the feasibility of implementing the cloning economically, i.e., without an ancilla, and determine the dimension of the ancilla when an economic implementation is not possible.
While the no-cloning theorem, which forbids the perfect copying of quantum states, is well-known as one of the defining features of quantum mechanics, the question of how well the theory allows a state to be cloned is yet to be completely solved. In this paper, rigorous solutions to the problem of M→N asymmetric cloning of qudits are obtained in a number of interesting cases. The central result is the solution to the 1→N universal asymmetric qudit cloning problem for which the exact trade-off in the fidelities of the clones for every N and d is derived. Analogous results are proven for qubits when M=N−1. We also consider state-dependent 1→N qubit cloning, providing a general parametrization in terms of a Heisenberg star Hamiltonian. In all instances, we determine the feasibility of implementing the cloning economically, i.e., without an ancilla, and determine the dimension of the ancilla when an economic implementation is not possible.
The possibility of cloning a d-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 d-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=2. The used ansatz is supported by numerical evidence up to d=7. An economical phase-covariant cloner can nevertheless be constructed for d>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=2.
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.