In this work, we introduce a special kind of quantum cloning machine called Hybrid quantum cloning machine. The introduced Hybrid quantum cloning machine or transformation is nothing but a combination of pre-existing quantum cloning transformations. In this sense it creates its own identity in the field of quantum cloners. Hybrid quantum cloning machine can be of two types: (i) State dependent and (ii) State independent or Universal. We study here the above two types of Hybrid quantum cloning machines. Later we will show that the state dependent hybrid quantum-cloning machine can be applied on only four input states. We will also find in this paper another asymmetric universal quantum cloning machine constructed from the combination of optimal universal B-H quantum cloning machine and universal anti-cloning machine. The fidelities of the two outputs are different and their values lie in the neighborhood of ${5/6} $
Possibility of state cloning is analyzed in two types of generalizations of quantum mechanics with nonlinear evolution. It is first shown that nonlinear Hamiltonian quantum mechanics does not admit cloning without the cloning machine. It is then demonstrated that the addition of the cloning machine, treated as a quantum or as a classical system, makes the cloning possible by nonlinear Hamiltonian evolution. However, a special type of quantum-classical theory, known as the mean-field Hamiltonian hybrid mechanics, does not admit cloning by natural evolution. The latter represents an example of a…
The no-cloning theorem states that an unknown quantum state cannot be cloned exactly and deterministically due to the linearity of quantum mechanics. Associated with this theorem is the quantitative no-cloning limit that sets an upper bound to the quality of the generated clones. However, this limit can be circumvented by abandoning determinism and using probabilistic methods. Here, we report an experimental demonstration of probabilistic cloning of arbitrary coherent states that clearly surpasses the no-cloning limit. Our scheme is based on a hybrid linear amplifier that combines an ideal det…
The trade-offs among various output fidelities of asymmetric universal cloning machines are investigated. First we find out all the attainable optimal output fidelities for the 1 to 3 asymmetric universal cloning machine and it turns out that there are two kinds of extremal asymmetric cloning machines which have to cooperate in order to achieve some of the optimal output fidelities. Second we construct a family of extremal cloning machines that includes the universal symmetric cloning machine as well as an asymmetric 1 to $1+N$ cloning machine for qudits with two different output fidelities su…
The optimal N to M ($M>N$) quantum cloning machines for the d-level system are presented. The unitary cloning transformations achieve the bound of the fidelity.
Quantum cloning machines for equatorial qubits are studied. For the case of 1 to 2 phase-covariant quantum cloning machine, we present the networks consisting of quantum gates to realize the quantum cloning transformations. The copied equatorial qubits are shown to be separable by using Peres-Horodecki criterion. The optimal 1 to M phase-covariant quantum cloning transformations are given.
We propose an optical implementation of the Gaussian continuous-variable quantum cloning machines. We construct a symmetric N -> M cloner which optimally clones coherent states and we also provide an explicit design of an asymmetric 1 -> 2 cloning machine. All proposed cloning devices can be built from just a single non-degenerate optical parametric amplifier and several beam splitters.
No-cloning theorem is fundamental for quantum mechanics and for quantum information science that states an unknown quantum state cannot be cloned perfectly. However, we can try to clone a quantum state approximately with the optimal fidelity, or instead, we can try to clone it perfectly with the largest probability. Thus various quantum cloning machines have been designed for different quantum information protocols. Specifically, quantum cloning machines can be designed to analyze the security of quantum key distribution protocols such as BB84 protocol, six-state protocol, B92 protocol and the…
We present a unified universal quantum cloning machine, which combines several different existing universal cloning machines together including the asymmetric case. In this unified framework, the identical pure states are projected equally into each copy initially constituted by input and one half of the maximally entangled states. We show explicitly that the output states of those universal cloning machines are the same. One importance of this unified cloning machine is that the cloning procession is always the symmetric projection which reduces dramatically the difficulties for implementatio…
Quantum cloning machine for arbitrary mixed states in symmetric subspace is proposed. This quantum cloning machine can be used to copy part of the output state of another quantum cloning machine and is useful in quantum computation and quantum information. The shrinking factor of this quantum cloning achieves the well-known upper bound. When the input is identical pure states, two different fidelities of this cloning machine are optimal.
We show that an unknown quantum state in phase space can be teleported via three-mode entanglement generated by continuous variable quantum cloning machine (transformation). Further, proceeding with our teleportation protocol we are able to improve the fidelity of teleportation obtained by Loock et.al. [Phys.Rev.Lett. 84, 3482(2000)]. Also we study here the entanglement between the two output copies from cloning machine.
An unknown quantum state cannot be copied on demand and broadcast freely due to the famous 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, roughly the same as that of fidelity. It is perhaps also unsurp…
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.
The universal quantum cloning machine and the universal NOT gate acting on a single qubit can be implemented very generally by slightly modifying the protocol of quantum state teleportation. The experimental demonstration of the 1 to 2 cloning process according to the above scheme has been realized for a qubit encoded in photon polarization.
After a brief introduction to the quantum no-cloning theorem and its link with the linearity and causality of quantum mechanics, the concept of quantum cloning machines is sketched, following, whenever possible, the chronology of the main results. The important classes of quantum cloning machines are reviewed, in particular state-independent and state-dependent cloning machines. The 1-to-2 cloning problem is then studied from a formal point of view, using the isomorphism between completely positive maps and operators, which leads to the so-called double-Bell ansatz. This also yields an efficie…
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 universa…
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…
we envisage a novel quantum cloning machine, which takes an input state and produces an output state whose success branch can exist in a linear superposition of multiple copies of the input state and the failure branch exist in a superposition of composite state independent of the input state. We prove that unknown non-orthogonal states chosen from a set $\cal S$ can evolve into a linear superposition of multiple clones by a unitary process if and only if the states are linearly independent. We derive a bound on the success probability of the novel cloning machine. We argue that the determinis…
Probabilistic cloning was first proposed by Duan and Guo. Then Pati established a novel cloning machine (NCM) for copying superposition of multiple clones simultaneously. In this paper, we deal with the novel cloning machine with supplementary information (NCMSI). For the case of cloning two states, we demonstrate that the optimal efficiency of the NCMSI in which the original party and the supplementary party can perform quantum communication equals that achieved by a two-step cloning protocol wherein classical communication is only allowed between the original and the supplementary parties. F…
A quantum cloning machine is introduced that yields $M$ identical optimal clones from $N$ replicas of a coherent state and $N'$ replicas of its phase conjugate. It also optimally produces $M'=M+N'-N$ phase-conjugated clones at no cost. For well chosen input asymmetries $N'/(N+N')$, this machine is shown to provide better cloning fidelities than the standard $(N+N') \to M$ cloner. The special cases of the optimal balanced cloner ($N=N'$) and the optimal measurement ($M=\infty$) are investigated.
Coherence, being at the heart of interference phenomena, is found to be an useful resource in quantum information theory. Here we want to understand quantum coherence under the combination of two fundamentally dual processes, viz., cloning and deleting. We found the role of quantum cloning and deletion machines with the consumption and generation of quantum coherence. We establish cloning as a cohering process and deletion as a decohering process. Fidelity of the process will be shown to have connection with coherence generation and consumption of the processes.
We apply a general method for the estimation of completely positive maps to the 1-to-2 universal covariant cloning machine. The method is based on the maximum-likelihood principle, and makes use of random input states, along with random projective measurements on the output clones. The downhill simplex algorithm is applied for the maximisation of the likelihood functional.
A family of asymmetric cloning machines for $N$-dimensional quantum states is introduced. These machines produce two imperfect copies of a single state that emerge from two distinct Heisenberg channels. The tradeoff between the quality of these copies is shown to result from a complementarity akin to Heisenberg uncertainty principle. A no-cloning inequality is derived for isotropic cloners: if $π_a$ and $π_b$ are the depolarizing fractions associated with the two copies, the domain in $(\sqrt{π_a},\sqrt{π_b})$-space located inside a particular ellipse representing close-to-perfect cloning is f…
The 1->3 quantum phase covariant cloning, which optimally clones qubits belonging to the equatorial plane of the Bloch sphere, achieves the fidelity Fcov(1->3)=0.833, larger than for the 1->3 universal cloning Funiv(1->3)=0.778. We show how the 1->3 phase covariant cloning can be implemented by a smart modification of the standard universal quantum machine by a projection of the output states over the symmetric subspace. A complete experimental realization of the protocol for polarization encoded qubits based on non-linear and linear methods will be discussed.
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 answer to this question and construct their famous B-H quantum cloning machine. The B-H machine clones state of an arbitrary single qubit in optimal manner and hence it is universal. Generalizing this machine for two-qubit system is straightforward, but this procedure does not preserve quantum correlation existing in bipartite state in optimal manner and also, during …
We present Quantum Cloning Machines (QCM) that transform N identical qubits into $M>N$ identical copies and we prove that the fidelity (quality) of these copies is optimal. The connection between cloning and measurement is discussed in detail. When the number of clones M tends towards infinity, the fidelity of each clone tends towards the optimal fidelity that can be obtained by a measurement on the input qubits. More generally, the QCM are universal devices to translate quantum information into classical information.
Recently Galvão and Hardy have shown that quantum cloning can improve the performance of some quantum computation tasks. However such performance enhancement is possible only if quantum correlations survive the cloning process. We investigate preservation of the quantum correlations in the process of non--local cloning of entangled pairs of two--level systems. We consider different kinds of quantum cloning machines and compare their effectiveness in the cloning of non--maximally entangled pure states. A mean entanglement is introduced in order to obtain a quantitative evaluation of an average …
Here we report an experimental realization of optimal phase-covariant quantum cloning machine with a single electron spin in solid state system at room temperature. The involved three states of two logic qubits are encoded physically in three levels of a single electron spin with two Zeeman sub-levels at a nitrogen-vacancy defect center in diamond. The preparation of input state and the phase-covariant quantum cloning transformation are controlled by two independent microwave fields. The average experimental fidelity reaches 85.2% which is very close to theoretical optimal fidelity 85.4% and i…
We propose two feasible experimental implementations of an optimal asymmetric 1->2 quantum cloning of a polarization state of photon. Both implementations are based on a partial and optimal reverse of recent conditional symmetrical quantum cloning experiments. The reversion procedure is performed only by a local measurement of one from the clones and ancilla followed by a local operation on the other clone. The local measurement consists only of a single unbalanced beam splitter followed in one output by a single photon detector and the asymmetry of fidelities in the cloning is controlled by a…
A general multi-step N->M probabilistic optimal universal cloning protocol is presented together with the experimental realization of the (1 -> 3) and (2 -> 3) machines. Since the present method exploits the bosonic nature of the photons, it can be applied to any particle obeying to the Bose statistics. On a technological perspective, the present protocol is expected to find applications as a novel, multi-qubit symmetrizator device to be used in modern quantum information networks.
The optimal phase covariant cloning machine (PQCM) broadcasts the information associated to an input qubit into a multi-qubit systems, exploiting a partial a-priori knowledge of the input state. This additional a priori information leads to a higher fidelity than for the universal cloning. The present article first analyzes different experimental schemes to implement the 1->3 PQCM. The method is then generalized to any 1->M machine for odd value of M by a theoretical approach based on the general angular momentum formalism. Finally different experimental schemes based either on linear or non-l…
We propose a scheme to implement the $1\to2$ universal quantum cloning machine of Buzek et.al [Phys. Rev.A 54, 1844(1996)] in the context of cavity QED. The scheme requires cavity-assisted collision processes between atoms, which cross through nonresonant cavity fields in the vacuum states. The cavity fields are only virtually excited to face the decoherence problem. That's why the requirements on the cavity quality factor can be loosened.
We analyze a possibility of using the two qubit output state from Buzek-Hillery quantum copying machine (not necessarily universal quantum cloning machine) as a teleportation channel. We show that there is a range of values of the machine parameter $ξ$ for which the two qubit output state is entangled and violates Bell-CHSH inequality and for a different range it remains entangled but does not violate Bell-CHSH inequality. Further we observe that for certain values of the machine parameter the two-qubit mixed state can be used as a teleportation channel. The use of the output state from the Bu…
In probabilistic cloning with two auxiliary systems, we consider and compare three different protocols for the success probabilities of cloning. We show that, in certain circumstances, it may increase the success probability to add an auxiliary system to the probabilistic cloning machine having one auxiliary system, but we always can find another cloning machine with one auxiliary system having the same success probability as that with two auxiliary systems.
We investigate the problem of cloning a set of states that is invariant under the action of an irreducible group representation. We then characterize the cloners that are "extremal" in the convex set of group covariant cloning machines, among which one can restrict the search for optimal cloners. For a set of states that is invariant under the discrete Weyl-Heisenberg group, we show that all extremal cloners can be unitarily realized using the so-called "double-Bell states", whence providing a general proof of the popular ansatz used in the literature for finding optimal cloners in a variety o…
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 illust…
We propose a physically realizable machine which can either generate multiparticle W-like states, or implement high fidelity $1 \to M$ ($M=1,2,... \infty$) anti-cloning of an arbitrary qubit state, in a single step. Moreover this universal machine acts as a catalyst in that it is unchanged after either procedure, effectively resetting itself for its next operation. It also possesses an inherent {\em immunity} to decoherence. Most importantly in terms of practical multi-party quantum communication, the machine's robustness in the presence of decoherence actually {\em increases} as the number of…
A simultaneous realization of the Universal Optimal Quantum Cloning Machine (UOQCM) and of the Universal-NOT gate by a quantum injected optical parametric amplification (QIOPA), is reported. The two processes, forbidden in their exact form for fundamental quantum limitations, are found universal and optimal, and the measured fidelity F<1 is found close to the limit values evaluated by quantum theory. This work may enlighten the yet little explored interconnections of fundamental axiomatic properties within the deep structure of quantum mechanics.
By a significant modification of the standard protocol of quantum state Teleportation two processes ''forbidden'' by quantum mechanics in their exact form, the Universal NOT gate and the Universal Optimal Quantum Cloning Machine, have been implemented contextually and optimally by a fully linear method. In particular, the first experimental demonstration of the Tele-UNOT Gate, a novel quantum information protocol has been reported (cfr. quant-ph/0304070). A complete experimental realization of the protocol is presented here.
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 arbitrary input qubit equally well, however can be determined completely by only four input states located at the four vertices of a tetrahedron. The phase-covariant quantum cloning machine, which can copy all qubits located on the equator of the Bloch sphere, can be determined by three equatorial qubits with equal angular distance. These results sharpen further the well-known results that BB84 states a…
The possibility of cloning a d-dimensional quantum system without an ancilla is explored, extending on the economical phase-covariant cloning machine found in [Phys. Rev. A {\bf 60}, 2764 (1999)] for qubits. We prove the impossibility of constructing an economical version of the optimal universal cloning machine in any dimension. We also show, using an ansatz on the generic form of cloning machines, that the d-dimensional phase-covariant cloner, which optimally clones all uniform superpositions, can be realized economically only in dimension d=2. The used ansatz is supported by numerical evide…
The impossibility of perfectly copying (or cloning) an arbitrary quantum state is one of the basic rules governing the physics of quantum systems. The processes that perform the optimal approximate cloning have been found in many cases. These "quantum cloning machines" are important tools for studying a wide variety of tasks, e.g. state estimation and eavesdropping on quantum cryptography. This paper provides a comprehensive review of quantum cloning machines (both for discrete-dimensional and for continuous-variable quantum systems); in addition, it presents the role of cloning in quantum cry…
In this paper, we study the entanglement and quantum discord of the output modes in the unified $1\rightarrow 2$ state-dependent cloning and probabilistic quantum cloning. The tripartite entanglement among the output modes and the quantum cloning machine is also considered. We find that the roles of the quantum correlations including the bipartite and tripartite entanglement and quantum discord strongly depend on the quantum cloning machines as well as the cloned state. In particular, it is found that this quantum cloning scheme can be realizable even without any quantum correlation.
We consider one of the quantum key distribution protocols recently introduced in Ref. [Pirandola et al., Nature Physics 4, 726 (2008)]. This protocol consists in a two-way quantum communication between Alice and Bob, where Alice encodes secret information via a random phase-space displacement of a coherent state. In particular, we study its security against a specific class of individual attacks which are based on combinations of Gaussian quantum cloning machines.
After the appearance of the no-cloning theorem, approximate quantum cloning machines (QCMs) have become one of the most 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 semi-definite 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…
A symmetric 1 to 2 quantum cloning machine (QCM) is presented that provides high-fidelity copies with $0.90 \le F \le 0.95$ for all pure (single-qubit) input states from a given meridian of the Bloch sphere. \cor{Emphasize is placed especially on the states of the (so-called) Eastern meridian, that includes the computational basis states $\ketm{0}, \ketm{1}$ together with the diagonal state $\ketm{+} = \frac{1}{\sqrt{2}} (\ketm{0} + \ketm{1})$, for which suggested cloning transformation is shown to be optimal.} In addition, we also show how this QCM can be utilized for eavesdropping in Bennett…
We discuss the "partial" quantum cloning of the pure two-partite states, when the "part" of initial state related to the one qubit is copied only. The same approach gives the possibility to design the quantum copying machine for the mixed qubit states.
The study of quantum cryptography and quantum entanglement has traditionally been based on two-level quantum systems (qubits) and more recently on three-level systems (qutrits). We investigate several classes of state-dependent quantum cloners for four-level systems (quartits). These results apply to symmetric as well as asymmetric cloners, so that the balance between the fidelity of the two clones can also be analyzed. We extend Cerf's formalism for cloning states in order to derive cloning machines that remain invariant under certain unitary transformations. Our results show that a different…
The cloning of continuous quantum variables is analyzed based on the concept of Gaussian cloning machines, i.e., transformations that yield copies that are Gaussian mixtures centered on the state to be copied. The optimality of Gaussian cloning machines that transform N identical input states into M output states is investigated, and bounds on the fidelity of the process are derived via a connection with quantum estimation theory. In particular, the optimal N-to-M cloning fidelity for coherent states is found to be equal to MN/(MN+M-N).
In this work, we design a deleting machine and shown that for some given condition on machine parameters, it gives slightly better result than P-B deleting machine [5,6]. Also it is shown that for some particular values of the machine parameters it acts like Pati-Braunstein deleting machine. We also study the combined effect of cloning and deleting machine, where at first the cloning is done by some standard cloning machines such as Wootters-Zurek [1] and Buzek-Hillery [2] cloning machine and then the copy mode is deleted by Pati-Braunstein deleting machine or our prescribed deleting machine. …
It is shown that any quantum operation that perfectly clones the entanglement of all maximally-entangled qubit pairs cannot preserve separability. This ``entanglement no-cloning'' principle naturally suggests that some approximate cloning of entanglement is nevertheless allowed by quantum mechanics. We investigate a separability-preserving optimal cloning machine that duplicates all maximally-entangled states of two qubits, resulting in 0.285 bits of entanglement per clone, while a local cloning machine only yields 0.060 bits of entanglement per clone.
We investigate the optimal distribution of quantum information over multipartite systems in asymmetric settings. We introduce cloning transformations that take $N$ identical replicas of a pure state in any dimension as input, and yield a collection of clones with non-identical fidelities. As an example, if the clones are partitioned into a set of $M_A$ clones with fidelity $F^A$ and another set of $M_B$ clones with fidelity $F^B$, the trade-off between these fidelities is analyzed, and particular cases of optimal $N \to M_A+M_B$ cloning machines are exhibited. We also present an optimal $1 \to…
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 $1\rightarrow 2$ qubits cloning. We then describe a different and new state dependent cloning protocol with four machine…
We study machines that take N identical replicas of a pure qudit state as input and output a set of M_A clones of a given fidelity and another set of $M_B$ clones of another fidelity. The trade-off between these two fidelities is investigated, and numerous examples of optimal N -> M_A+M_B cloning machines are exhibited using a generic method. A generalisation to more than two sets of clones is also discussed. Finally, an optical implementation of some such machines is proposed. This paper is an extended version of [xxx.arxiv.org/abs/quant-ph/0411179].
Beyond the no-cloning theorem, the universal symmetric quantum cloning machine was first addressed by Buzek and Hillery. Here, we realized the one-to-two qubits Buzek-Hillery cloning machine with linear optical devices. This method relies on the representation of several qubits by a single photon. We showed that, the fidelities between the two output qubits and the original qubit are both 5/6 (which proved to be the optimal fidelity of one-to-two qubits universal cloner) for arbitrary input pure states.
We present a feasible scheme to use trapped ions Cavity QED system to implement optimal $1\to N$ cloning machine of coherent state. In present scheme, as the ouput of the clone machine, the copies of the cavity mode emerge at the vibrational modes and cavity modes is acted as the ancilla mode in the implementation of the cloning. We further show that such system can be used to generated multimodes quantum state, which can realize optimum symmetric $1\to N$ telecloning of coherent state of distant cavity.
We investigate a new 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.
It is well known that it is impossible to clone an arbitrary quantum state. However, this inability does not lead directly to no-cloning of quantum coherence. Here, we show that it is impossible to clone the coherence of an arbitrary quantum state which is a stronger statement than the 'no-cloning of quantum state'. In particular, with ancillary system as machine state, we show that it is impossible to clone the coherence of states whose coherence is greater than the coherence of the known states on which the transformations are defined. Also, we characterize the class of states for which cohe…
We describe a nuclear magnetic resonance (NMR) experiment which implements an efficient one-to-two qubit phase-covariant cloning machine(QPCCM). In the experiment we have achieved remarkably high fidelities of cloning, 0.848 and 0.844 respectively for the original and the blank qubit. This experimental value is close to the optimal theoretical value of 0.854. We have also demonstrated how to use our phase-covariant cloning machine for quantum simulations of bit by bit eavesdropping in the four-state quantum key distribution protocol.
