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Physical Review A

Buzek-Hillery cloning revisited using the bures metric and trace norm

L. C. Kwek, C. H. Oh, Xiang‐Bin Wang, Yee‐Chia Yeo

Published Oct 18, 2000• 13 citations
Publisher

Abstract

Buz\ifmmode \breve{}\else \u{}\fi{}ek and Hillery have recently analyzed the possibility of cloning imperfectly arbitrary states using a universal quantum cloning machine. They have analyzed their result using the Hilbert-Schmidt norm, although a better measure of fidelity is the Bures metric. In this paper, we repeat the Buz\ifmmode \breve{}\else \u{}\fi{}ek and Hillery result using the Bures metric and show that we can still obtain the same result with an improved measure. However, unlike the Hilbert-Schmidt norm, this computation may be extended to the case of continuous variable, infinite-dimensional spaces, or even Hilbert spaces of larger dimensions. For completeness, a similar analysis using the trace norm is also performed.

DOI
10.1103/physreva.62.052313

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Buz\ifmmode \breve{}\else \u{}\fi{}ek and Hillery have recently analyzed the possibility of cloning imperfectly arbitrary states using a universal quantum cloning machine. They have analyzed their result using the Hilbert-Schmidt norm, although a better measure of fidelity is the Bures metric. In this paper, we repeat the Buz\ifmmode \breve{}\else \u{}\fi{}ek and Hillery result using the Bures metric and show that we can still obtain the same result with an improved measure. However, unlike the Hilbert-Schmidt norm, this computation may be extended to the case of continuous variable, infinite-dimensional spaces, or even Hilbert spaces of larger dimensions. For completeness, a similar analysis using the trace norm is also performed.
Bužek and Hillery proposed a universal quantum-copying machine (UQCM) (i.e., transformation) to analyze the possibility of cloning arbitrary states. The UQCM copies quantum-mechanical states with the quality of its output does not depend on the input. We propose a slightly different transformation to analyze a restricted set of input states. We impose the conditions (I) the density matrices of the two output states are the same, and that (II) the distance between input density operator and the output density operators is input state independent. Using Hilbert-Schmidt norm and Bures fidelity, we show that our transformation can achieves the bound of the fidelity.
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.
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.
Bužek and M. Hillery analyze the possibility of (e) copying (that is, cloning) arbitrary states of (e) a quantum-mechanical spin-1/2 system. Authors show that there exists (eb) universal quantum-copying machine‘‘ (i.e., transformation) which approximately copies quantum-mechanical states such that (e) the quality of its output does not (e) depend on the input. Authors also examine a machine which combines a unitary transformation and (e&eb) a selective measurement to produce good copies of (e) states in the neighborhood of a particular state. Authors discuss the problem of (e) measurement of the output states. © 1996 The American Physical Society Received 5 February 1996 DOI:http://dx.doi.org/10.1103/PhysRevA.54.1844 Quantum copying: Beyond the no-cloning theorem V. Bužek and M. Hillery Phys. Rev. A 54, 1844 – Published 1 September 1996 STOP Authors derive a necessary and sufficient condition for two pure states, each entangled in two remote systems, to be clonable by the sequential access to the two systems. The result shows (eb) to what extent... The full paper: http://www.iiste.org/PDFshare/APTA-PAGENO-496948-501458.pdf
Due to the axioms of quantum mechanics, perfect cloning of an unknown quantum state is impossible. But since imperfect cloning is still possible, a question arises: ``Is there an optimal quantum cloning machine?'' Buzek and Hillery answered this question and constructed their famous B-H quantum cloning machine. The B-H machine clones the state of an arbitrary single qubit in an optimal manner and hence it is universal. Generalizing this machine for a two-qubit system is straightforward, but during this procedure, except for product states, this machine loses its universality and becomes a state-dependent cloning machine. In this paper, we propose some classes of optimal universal local quantum state cloners for a particular class of two-qubit systems, more precisely, for a class of states with known Schmidt basis. We then extend our machine to the case that the Schmidt basis of the input state is deviated from the local computational basis of the machine. We show that more local quantum coherence existing in the input state corresponds to less fidelity between the input and output states. Also we present two classes of a state-dependent local quantum copying machine. Furthermore, we investigate local broadcasting of two aspects of quantum correlations, i.e., quantum entanglement and quantum discord, defined, respectively, within the entanglement-separability paradigm and from an information-theoretic perspective. The results show that although quantum correlation is, in general, very fragile during the broadcasting procedure, quantum discord is broadcasted more robustly than quantum entanglement.