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Journal of Physics A Mathematical and General

Phase-covariant quantum cloning

Heng Fan, Keiji Matsumoto, Xiang‐Bin Wang, Hiroshi Imai

Published Aug 13, 2002• 13 citations
Publisher

Abstract

Quantum cloning machines for equatorial qubits are studied. For a 1 to 2 phase-covariant quantum cloning machine, using Hilbert–Schmidt norm and Bures fidelity, we show that our transformation can achieve the bound of the fidelity.
DOI
10.1088/0305-4470/35/34/312

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Since the initial discovery of the Wootters-Zurek no-cloning theorem, a wide variety of quantum cloning machines have been proposed aiming at imperfect but optimal cloning of quantum states within its own context. Remarkably, most previous studies have employed the Bures fidelity or the Hilbert-Schmidt norm as the figure of merit to characterize the quality of the corresponding cloning scenarios. However, in many situations, what we truly care about is the relevant information about certain parameters encoded in quantum states. In this work, we investigate the multiple phase estimation problem in the framework of quantum cloning machines, from the perspective of quantum Fisher information matrix (QFIM). Focusing on the generalized ddd-dimensional equatorial states, we obtain the analytical formulas of QFIM for both universal quantum cloning machine (UQCM) and phase-covariant quantum cloning machine (PQCM), and prove that PQCM indeed performs better than UQCM in terms of QFIM. We highlight that our method can be generalized to arbitrary cloning schemes where the fidelity between the single-copy input and output states is input-state independent. Furthermore, the attainability of the quantum Cram\'er-Rao bound is also explicitly discussed.
It is known that in phase-covariant quantum cloning, the equatorial states on the Bloch sphere can be cloned with a fidelity higher than the optimal bound established for universal quantum cloning. We generalize this concept to include other states on the Bloch sphere with a definite z component of spin. It is shown that once we know the z component, we can always clone a state with a fidelity higher than the universal value and that of equatorial states. We also make a detailed study of the entanglement properties of the output copies and show that the equatorial states are the only states that give rise to a separable density matrix for the outputs.
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 ddd with M=kd+NM=kd+NM=kd+N, kkk 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.
Maximum possible fidelity in 1→21\rightarrow 21→2 qubits cloning
We re-analyse the Bužek-Hillery Universal Quantum Cloning machine protocol and show that it allows better values for fidelity and Hilbert-Schmidt norm than hitherto reported. This higher value for the fidelity is identical to the maximum fidelity of phase covariant quantum cloning of Bruß-Cinchetti-D'Ariano-Macchiavello. This value of fidelity has also been obtained by Niu and Griffiths in their work without machine states. This is the maximum possible fidelity obtainable in 1→21\rightarrow 21→2 qubits cloning. We then describe a different and new state dependent cloning protocol with four machine states where all non-exact copies of input states are taken into account in the output and we use the Hessian method of determining extrema of multivariate functions. The fidelity for the best overall quantum cloning in this protocol is Fˉ=0.847\bar{F}=0.847Fˉ=0.847 with an associated von-Neumann entropy of Sˉ=0.825\bar{S}=0.825Sˉ=0.825.