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arXiv:2107.03042 · Physical Review A

Process-optimized phase-covariant quantum cloning

Chloe Kim, Eric Chitambar

Published Aug 9, 2022• 1 citation
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Abstract

After the appearance of the no-cloning theorem, approximate quantum cloning machines (QCMs) has become a well-studied subject in quantum information theory. Among several measures to quantify the performance of a QCM, single-qudit fidelity and global fidelity have been most widely used. In this paper we compute the optimal global fidelity for phase-covariant cloning machines via semidefinite programming optimization, thereby completing a remaining gap in the previous results on QCMs. We also consider optimal simulations of the cloning and transpose cloning map, both by a direct optimization and by a composition of component-wise optimal QCMs. For the cloning map the composition method is suboptimal whereas for the transpose cloning map the method is asymptotically optimal.

DOI
10.1103/physreva.106.022405

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After the appearance of the no-cloning theorem, approximate quantum cloning machines (QCMs) has become a well-studied subject in quantum information theory. Among several measures to quantify the performance of a QCM, single-qudit fidelity and global fidelity have been most widely used. In this paper we compute the optimal global fidelity for phase-covariant cloning machines via semidefinite programming optimization, thereby completing a remaining gap in the previous results on QCMs. We also consider optimal simulations of the cloning and transpose cloning map, both by a direct optimization and by a composition of component-wise optimal QCMs. For the cloning map the composition method is suboptimal whereas for the transpose cloning map the method is asymptotically optimal.
Following the work of Niu and Griffiths, in \emph{Phys.Rev.A 58, 4377(1998)}, we shall investigate the problem, how to design the optimal quantum cloning machines (QCMs) for qubit system, with the help of Bloch-sphere representation. In stead of the quality factor there, the Fiuráusek's optimal condition, where the optimal cloning machine should maximize a convex mixture of the average fidelity, is used as the optimality criterion in present protocol. Almost all of the known optimal QCMs in previous works, the cloning for states with fixed polar angle, the phase-covariant cloning, the universal QCMs, the cloning for two arbitrary pure states, and the mirror phase-covariant cloning, should be discussed in a systematic way. The known results, the optimal fidelities for various input ensembles according to different optimality criteria, are recovered here. Our present scheme also offers a general way of constructing the unitary transformation to realize the optimal cloning.
While algebraic derivations establish theoretical limits for quantum cloning, practical implementations require explicit operator representations that are often unavailable analytically. We present a computational framework that reformulates cloning optimization as a search over completely positive trace-preserving maps using the Choi-Jamiolkowski isomorphism and Semidefinite Programming. The framework (i) numerically certifies global optimality through primal-dual strong duality and (ii) automatically extracts operational Kraus operators from the optimal Choi matrix via spectral decomposition. We systematically treat universal, phase-covariant, asymmetric, and entanglement cloning scenarios, providing -for the first time - a unified computational catalogue of explicit, implementable Kraus representations across all major cloning families, including higher-order processes and arbitrary input state distributions. As an application, we analyse optimal cloning attacks on BB84 under depolarizing noise, demonstrating how the extracted operators enable quantitative security analysis in realistic noisy quantum channels. An open-source implementation enables community validation and extension.
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.
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.
An unknown quantum state cannot be copied and broadcast freely due to the no-cloning theorem. Approximate cloning schemes have been proposed to achieve the optimal cloning characterized by the maximal fidelity between the original and its copies. Here, from the perspective of quantum Fisher information (QFI), we investigate the distribution of QFI in asymmetric cloning machines which produce two nonidentical copies. As one might expect, improving the QFI of one copy results in decreasing the QFI of the other copy. It is perhaps also unsurprising that asymmetric phase-covariant cloning outperforms universal cloning in distributing QFI since a priori information of the input state has been utilized. However, interesting results appear when we compare the distributabilities of fidelity (which quantifies the full information of quantum states), and QFI (which only captures the information of relevant parameters) in asymmetric cloning machines. Unlike the results of fidelity, where the distributability of symmetric cloning is always optimal for any d-dimensional cloning, we find that any asymmetric cloning outperforms symmetric cloning on the distribution of QFI for d ≤ 18, whereas some but not all asymmetric cloning strategies could be worse than symmetric ones when d > 18.
We find an optimal quantum cloning machine, which clones qubits of arbitrary symmetrical distribution around the Bloch vector with the highest fidelity. The process is referred to as phase-independent cloning in contrast to the standard phase-covariant cloning for which an input qubit state is a priori better known. We assume that the information about the input state is encoded in an arbitrary axisymmetric distribution (phase function) on the Bloch sphere of the cloned qubits. We find analytical expressions describing the optimal cloning transformation and fidelity of the clones. As an illustration, we analyze cloning of qubit state described by the von Mises--Fisher and Brosseau distributions. Moreover, we show that the optimal phase-independent cloning machine can be implemented by modifying the mirror phase-covariant cloning machine for which quantum circuits are known.