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arXiv:1208.0746 · Physical Review A

Minimal input sets determining phase-covariant and universal quantum cloning

Jing Li, Yi-Nan Wang, Handuo Shi, Liang-Zhu Mu, Heng Fan

Published Dec 18, 2012• 9 citations
arXivHTMLPDF

Abstract

We study the minimal input sets which can determine completely the universal and the phase-covariant quantum cloning machines. We find that the universal quantum cloning machine, which can copy an arbitrary input qubit to two identical copies, however, can be determined completely by only four input states located at the four vertices of a tetrahedron in a Bloch sphere. The phase-covariant quantum cloning machine, which can create two copies from an arbitrary qubit located on the equator of the Bloch sphere, can be determined by three qubits located symmetrically on the equator of the Bloch sphere with equal relative phase. These results sharpen further the well-known results that Bennett-Brassard 1984 protocol (BB84) states and six states used in quantum cryptography can determine completely the phase-covariant and universal quantum cloning machines. This can simplify the testing procedure of whether the quantum clone machines are successful or not; namely, we only need to check that the minimal input sets can be cloned optimally, which can ensure that the quantum clone machines can work well for all input states.

DOI
10.1103/physreva.86.062315

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We study the minimal input sets which can determine completely the universal and the phase-covariant quantum cloning machines. We find that the universal quantum cloning machine, which can copy an arbitrary input qubit to two identical copies, however, can be determined completely by only four input states located at the four vertices of a tetrahedron in a Bloch sphere. The phase-covariant quantum cloning machine, which can create two copies from an arbitrary qubit located on the equator of the Bloch sphere, can be determined by three qubits located symmetrically on the equator of the Bloch sphere with equal relative phase. These results sharpen further the well-known results that Bennett-Brassard 1984 protocol (BB84) states and six states used in quantum cryptography can determine completely the phase-covariant and universal quantum cloning machines. This can simplify the testing procedure of whether the quantum clone machines are successful or not; namely, we only need to check that the minimal input sets can be cloned optimally, which can ensure that the quantum clone machines can work well for all input states.
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.
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 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.
We experimentally realize the $1\ensuremath{\rightarrow}2$ optimal universal quantum cloning machine for arbitrary qubits and the ancilla-free $1\ensuremath{\rightarrow}3$ optimal phase-covariant quantum cloning machine for $x\text{\ensuremath{-}}y$ equatorial qubits, which were constructed by Zou et al. [Phys. Lett. A 366, 36 (2007)]. The qubits are represented by the polarization states of photons prepared from the process of spontaneous parametric down-conversion. The copies of each machine are shown to have almost equal fidelities conditioned on fourfold coincidences.
Abstract In quantum mechanics, although the no‐cloning theorem forbids us to clone an arbitrary quantum state ideally, it is still possible to partially reproduce an unknown quantum state by using a phase cloning machine. Since the no‐cloning theorem guarantees the safety of quantum key distribution (QKD), resulting in the famous Bennett–Brassard 1984 (BB84) protocol, phase cloning of BB84 states increases the probability to eavesdrop the transmitted information between correspondents, forcing us to recheck the security of the QKD protocol. In this work, we realize an optimal phase cloning machine for BB84 states in a superconducting quantum bit (qubit). The lowest three energy levels in this system are exploited as logical qubits to demonstrate this cloning procedure. Two microwaves are used to prepare and clone the states. In order to achieve the maximum fidelity, we use the state tomography to readout the experimental results. It is found that the state information is well preserved during this cloning process. The average fidelity between input and output states is 83.92%, making this a valid quantum cloning machine.
We study state-dependent quantum cloning that can outperform universal cloning (UC).This is possible by using some a priori information on a given quantum state to be cloned.Specifically, we propose a generalization and optical implementation of quantum optimal mirror phase-covariant cloning, which refers to optimal cloning of sets of qubits of known modulus of the expectation value of Pauli's Z operator.Our results can be applied to cloning of an arbitrary mirror-symmetric distribution of qubits on the Bloch sphere including in special cases UC and phase-covariant cloning.We show that the cloning is optimal by adapting our former optimality proof for axisymmetric cloning (Bartkiewicz and Miranowicz 2010 Phys.Rev. A 82 042330).Moreover, we propose an optical realization of the optimal mirror phase-covariant 1 → 2 cloning of a qubit, for which the mean probability of successful cloning varies from 1/6 to 1/3 depending on prior information on the set of qubits to be cloned.The qubits are represented by polarization states of photons generated by the type-I spontaneous parametric down-conversion.The scheme is based on the interference of two photons on an unbalanced polarization-dependent beam splitter with different splitting ratios for vertical and horizontal polarization components and the additional application of feedforward by means of Pockels cells.The experimental feasibility of the proposed setup is carefully studied including various kinds of imperfections and losses.Moreover, we briefly describe two possible cryptographic applications of the optimal mirror phase-covariant cloning corresponding to state discrimination (or estimation) and secure quantum teleportation.