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.
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…
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…
After proving a general no-cloning theorem for black boxes, we derive the optimal universal cloning of unitary transformations, from one to two copies. The optimal cloner is realized by quantum channels with memory, and greately outperforms the optimal measure-and-reprepare cloning strategy. Applications are outlined, including two-way quantum cryptographic protocols.
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…
We study the optimal cloning transformation for two pairs of orthogonal states of two-dimensional quantum systems, and derive the corresponding optimal fidelities.
In this paper we present an approach to quantum cloning via free dynamical evolution of spin networks. By properly designing the network and the couplings between spins, we show that optimal 1->M phase covariant cloning can be achieved without any external control. Especially, when M is an odd number, the optimal phase-covariant cloning can be achieved without ancillas. Moreover, we demonstrate that the same framework is capable for optimal 1->2 universal cloning.
We show how to implement the optimal Gaussian $N$-to-$M$ cloning with linear optics and homodyne detection. We also show that the Gaussian $N$-to-$M$ cloning of known-phase coherent states can be performed with the fidelity $\sqrt \frac{2 M N}{2M N+M -N}$ by linear optics and homodyne detection, and with $\frac{2}{\sqrt{1+\frac{1}{N}}+\sqrt {1-\frac{1}{M}}}$ by utilizing quadrature squeezing. From the classical limit of the cloning (1-to-$\infty$ cloning), a necessary condition of continuous variable quantum key distribution using known-phase coherent states is provided.
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…
We consider non-universal cloning maps, namely cloning transformations which are covariant under a proper subgroup G of the universal unitary group U(d), where d is the dimension of the Hilbert space H of the system to be cloned. We give a general method for optimizing cloning for any cost-function. Examples of applications are given for the phase-covariant cloning (cloning of equatorial qubits) and for the Weyl-Heisenberg group (cloning of "continuous variables").
We derive optimal cloning limits for finite Gaussian distributions of coherent states, and describe techniques for achieving them. We discuss the relation of these limits to state estimation and the no-cloning limit in teleportation. A qualitatively different cloning limit is derived for a single-quadrature Gaussian quantum cloner.
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 show that optimal universal quantum cloning can be realized via stimulated emission. Universality of the cloning procedure is achieved by choosing systems that have appropriate symmetries. We first discuss a scheme based on stimulated emission in certain three-level-systems, e.g. atoms in a cavity. Then we present a way of realizing optimal universal cloning based on stimulated parametric down-conversion. This scheme also implements the optimal universal NOT operation.
We propose a quantum cloning machine, which clones a qubit into two clones assuming known modulus of expectation value of Pauli 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 Fiurasek [Phys. Rev. A 64, 062310 (2001)]. Quantum circuits implementing the optimal clon…
We investigate the asymmetric Gaussian cloning of coherent states which produces M copies from N input replicas, such that the fidelity of all copies may be different. We show that the optimal asymmetric Gaussian cloning can be performed with a single phase-insensitive amplifier and an array of beam splitters. We obtain a simple analytical expression characterizing the set of optimal asymmetric Gaussian cloning machines.
We consider cloning transformations of equatorial qubits and qutrits, with the transformation covariant for rotation of the phases. The optimal cloning maps are derived without simplifying assumptions from first principles, for any number of input and output qubits, and for a single input qutrit and any number of output qutrits. We also compare the cloning maps for global and single particle fidelities, and we show that the two criteria lead to different optimal maps.
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.
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…
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 …
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…
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).
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 transformation achieving the optimal symmetric N-to-M cloning of coherent states is presented. Its implementation only requires a phase-insensitive linear amplifier and a network of beam splitters. An experimental demonstration of this continuous-variable cloner should therefore be in the scope of current technology. The link between optimal quantum cloning and optimal amplification of quantum states is also pointed out.
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 investigate the security of continuous-variable (CV) quantum key distribution (QKD) using coherent states and reverse reconciliation against Gaussian individual attacks based on an optimal Gaussian $1 \to 2$ cloning machine. We provide an implementation of the optimal Gaussian individual attack. we also find a Bell-measurement attack which works without delayed choice of measurements and has better performance than the cloning attack.
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…
We propose a feasible scheme of conditional quantum partial teleportation of a qubit as optimal asymmetric cloning at a distance. In this scheme, Alice preserves one imperfect clone whereas other clone is teleported to Bob. Fidelities of the clones can be simply controlled by an asymmetry in Bell-state measurement. The optimality means that tightest inequality for the fidelities in the asymmetric cloning is saturated. Further we design a conditional teleportation as symmetric optimal N-> N+1 cloning from N Alice's replicas on single distant clone. We shortly discussed two feasible experimental…
The relative error of cloning of quantum states with arbitrary prior probabilities is considered. It is assumed that the ancilla may contain some a priori information about the input state to be cloned. The lower bound on the relative error for general cloning scenario is derived. Both the case of two-state set and case of multi-state set are analyzed in details. The treated figure of merit is compared with other optimality criteria. The quantum circuit for optimal cloning of a pair of pure states is constructed.
We describe an optical scheme for optimal Gaussian n to m cloning of coherent states. The scheme, which generalizes a recently demonstrated scheme for 1 to 2 cloning, involves only linear optical components and homodyne detection.
We construct the optimal 1 to 2 cloning transformation for the family of displaced thermal equilibrium states of a harmonic oscillator, with a fixed and known temperature. The transformation is Gaussian and it is optimal with respect to the figure of merit based on the joint output state and norm distance. The proof of the result is based on the equivalence between the optimal cloning problem and that of optimal amplification of Gaussian states which is then reduced to an optimization problem for diagonal states of a quantum oscillator. A key concept in finding the optimum is that of stochasti…
We show that universally covariant cloning is not optimal for achieving joint measurements of noncommuting observables with minimum added noise. For such a purpose a cloning transformation that is covariant with respect to a restricted transformation group is needed.
We propose a feasible scheme to implement the 1-to-2 optimal cloning transformation for two pairs of orthogonal states of two-dimensional quantum systems in the context of cavity QED. The copied qubits are shown to be inseparable by using Peres-Horodecki criterion.
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 construct the unique optimal quantum device for turning a finite number of d-level quantum systems in the same unknown pure state σinto M systems of the same kind, in an approximation of the M-fold tensor product of the state σ.
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…
We report an experimental realization of both optimal asymmetric cloning and telecloning of single photons by making use of partial teleportation of an unknown state. In the experiment, we demonstrate that, conditioned on the success of partial teleportation of single photons, not only the optimal asymmetric cloning can be accomplished, but also one of two outputs can be transfered to a distant location, realizing the telecloning. The experimental results represent a novel way to achieve the quantum cloning and may have potential applications in the context of quantum communication.
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…
