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arXiv:quant-ph/0301175 · Physical Review A

Optimal phase-covariant cloning for qubits and qutrits

Giacomo Mauro D’Ariano, Chiara Macchiavello

Published Apr 11, 2003• 104 citations

Abstract

We consider cloning transformations of equatorial qubits $|{\ensuremath{\psi}}_{\ensuremath{\varphi}}〉=1/\sqrt{2}(|0〉{+e}^{i\ensuremath{\varphi}}|1〉)$ and qutrits $|{\ensuremath{\psi}}_{\ensuremath{\varphi},\ensuremath{\theta}}〉=1/\sqrt{3}(|0〉{+e}^{i\ensuremath{\varphi}}|1〉{+e}^{i\ensuremath{\theta}}|2〉),$ with the transformation covariant for rotation of the phases $\ensuremath{\varphi}$ and $\ensuremath{\theta}.$ The optimal cloning maps are derived without simplifying assumptions from first principles, for any number of input and output qubits, and for a single-input qutrit and any number of output qutrits. We also compare the cloning maps for global and single-particle fidelities, and we show that the two criteria lead to different optimal maps.

DOI: 10.1103/physreva.67.042306

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We consider cloning transformations of equatorial qubits

|{\ensuremath{\psi}}_{\ensuremath{\varphi}}〉=1/\sqrt{2}(|0〉{+e}^{i\ensuremath{\varphi}}|1〉)

and qutrits

|{\ensuremath{\psi}}_{\ensuremath{\varphi},\ensuremath{\theta}}〉=1/\sqrt{3}(|0〉{+e}^{i\ensuremath{\varphi}}|1〉{+e}^{i\ensuremath{\theta}}|2〉),

with the transformation covariant for rotation of the phases

\ensuremath{\varphi}

and

\ensuremath{\theta}.

The optimal cloning maps are derived without simplifying assumptions from first principles, for any number of input and output qubits, and for a single-input qutrit and any number of output qutrits. We also compare the cloning maps for global and single-particle fidelities, and we show that the two criteria lead to different optimal maps.

We derive the optimal $N\ensuremath{\rightarrow}M$ phase-covariant quantum cloning for equatorial states in dimension

d

with

M=kd+N

,

k

integer. The cloning maps are optimal for both global and single-qudit fidelity. The map is achieved by an ``economical'' cloning machine, which works without ancilla.

We consider nonuniversal cloning maps, namely, cloning transformations that are covariant under a proper subgroup G of the universal unitary group

U(d),

where d is the dimension of the Hilbert space

\mathcal{H}

of the system to be cloned. We give a general method for optimizing cloning for any cost function. Examples of applications are given for phase-covariant cloning (cloning of equatorial qubits and qutrits) and for the Weyl-Heisenberg group (cloning of ``continuous variables'').

We propose a quantum cloning machine, which clones a qubit into two clones assuming known modulus of expectation value of Pauli ${\ensuremath{\sigma}}_{z}$ matrix. The process is referred to as the mirror phase-covariant cloning for which the input state is a priori less known than that for the standard phase-covariant cloning. Analytical expressions describing the cloning transformation and fidelity of the clones are found. Extremal equations for the optimal cloning are derived and analytically solved by generalizing a method of Fiur\'a\ifmmode \check{s}\else \v{s}\fi{}ek [Phys. Rev. A 64, 062310 (2001)]. Quantum circuits implementing the optimal cloning transformation and their physical realization in a quantum-dot system are described.

We consider cloning transformations of d-dimensional states of the form

e^{i\phi_0}|0> + e^{i\phi_1}|1> +...+ e^{i\phi_{d-1}}|d-1>

that are covariant with respect to rotations of the phases

\phi_i

's. The optimal cloning maps are easily obtained using a well-defined general characterization of state-dependent

1 \rightarrow 2

cloning transformations in arbitrary dimensions. Our results apply to symmetric as well as asymmetric cloners, so that the balance between the fidelity of the two clones can be analyzed.