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arXiv:quant-ph/0211169

Optimal asymmetric cloning machine on a great circle using no-signalling condition

Samir Kunkri, Md. Manirul Ali, Geetu Narang, Debasis Sarkar

Published Nov 26, 2002• 0 citations
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Abstract

Assuming the condition of no superluminal signalling, we got an upper bound on the quality of all asymmetric $ 1\to 2$ cloning machines, acting on qubits whose Bloch vectors lie on a great circle. Then we constructed an $ 1\to 2$ cloning machine, which asymmetrically clone all qubits corresponding to this great circle, and this machine matches with that upper bound, and hence this is optimal one.

DOI
10.48550/arxiv.quant-ph/0211169

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Assuming the condition of no superluminal signalling, we got an upper bound on the quality of all asymmetric 1→2 1\to 21→2 cloning machines, acting on qubits whose Bloch vectors lie on a great circle. Then we constructed an 1→2 1\to 21→2 cloning machine, which asymmetrically clone all qubits corresponding to this great circle, and this machine matches with that upper bound, and hence this is optimal one.
A family of asymmetric cloning machines for quantum bits and NNN-dimensional quantum states is introduced. These machines produce two approximate copies of a single quantum state that emerge from two distinct channels. In particular, an asymmetric Pauli cloning machine is defined that makes two imperfect copies of a quantum bit, while the overall input-to-output operation for each copy is a Pauli channel. A no-cloning inequality is derived, characterizing the impossibility of copying imposed by quantum mechanics. If ppp and p′p'p′ are the probabilities of the depolarizing channels associated with the two outputs, the domain in (sqrtp,sqrtp′)(\\sqrt{p},\\sqrt{p'})(sqrtp,sqrtp′)-space located inside a particular ellipse representing close-to-perfect cloning is forbidden. This ellipse tends to a circle when copying an NNN-dimensional state with NtoinftyN\\to\\inftyNtoinfty, which has a simple semi-classical interpretation. The symmetric Pauli cloning machines are then used to provide an upper bound on the quantum capacity of the Pauli channel of probabilities pxp_xpx​, pyp_ypy​ and pzp_zpz​. The capacity is proven to be vanishing if (sqrtpx,sqrtpy,sqrtpz)(\\sqrt{p_x},\\sqrt{p_y},\\sqrt{p_z})(sqrtpx​,sqrtpy​,sqrtpz​) lies outside an ellipsoid whose pole coincides with the depolarizing channel that underlies the universal cloning machine. Finally, the tradeoff between the quality of the two copies is shown to result from a complementarity akin to Heisenberg uncertainty principle.
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.
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.
While exact cloning of an unknown quantum state is prohibited by the linearity of quantum mechanics, approximate cloning is possible and has been used, e.g., to derive limits on the security of quantum communication protocols. In the case of asymmetric cloning, the information from the input state is distributed asymmetrically between the different output states. Here, we consider asymmetric phase-covariant cloning, where the goal is to optimally transfer the phase information from a single input qubit to different output qubits. We construct an optimal quantum cloning machine for two qubits that does not require ancilla qubits and implement it on an NMR quantum information processor.
We propose an optimal $1\ensuremath{\rightarrow}2$ quantum cloning method based on the maximin principle by making full use of a priori information of amplitude and phase about the general cloned qubit input set, which is a simply connected region enclosed by a ``longitude-latitude grid'' on the Bloch sphere. Theoretically, the fidelity of the optimal quantum cloning machine derived from this method is the largest in terms of the maximin principle compared with that of any other machine. The problem solving is an optimization process that involves six unknown complex variables, six vectors in an uncertain-dimensional complex vector space, and four equality constraints. Moreover, by restricting the structure of the quantum cloning machine, the optimization problem is simplified as a three-real-parameter suboptimization problem with only one equality constraint. We obtain the explicit formula for a suboptimal quantum cloning machine. Additionally, the fidelity of our suboptimal quantum cloning machine is higher than or at least equal to that of universal quantum cloning machines and phase-covariant quantum cloning machines. It is also underlined that the suboptimal cloning machine outperforms the ``belt quantum cloning machine'' for some cases.
We investigate the class of quantum cloning machines that equally duplicate all real states in a Hilbert space of arbitrary dimension. By using the no-signaling condition, namely, that cloning cannot make superluminal communication possible, we derive an upper bound on the fidelity of this class of quantum cloning machines. Then, for each dimension d, we construct an optimal symmetric cloner whose fidelity saturates this bound. Similar calculations can also be performed in order to recover the fidelity of the optimal universal cloner in d dimensions.