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.
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.
We consider optimal cloning of the spin coherent states in Hilbert spaces of different dimensionality d. We give explicit form of optimal cloning transformation for spin coherent states in the three-dimensional space, analytical results for the fidelity of the optimal cloning in d=3 and d=4 as well as numerical results for higher dimensions. In the low-dimensional case we construct the corresponding completely positive maps and exhibit their structure with the help of Jamiolkowski isomorphism. This allows us to formulate some conjectures about the form of optimal coherent cloning CP maps in ar…
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 give an alternative formulation of the no-cloning theorem that applies to harmonic oscillator coherent states. It says that {\em unknown} single harmonic oscillator coherent states can not be {\em amplified}. Conversely it says that {\em known} harmonic oscillator coherent states can be directly amplified through an universal unitary process. It is also shown that such a formulation is needed for the consistency of {\em information cloning} proposed by us long ago.
A scheme for optimal Gaussian cloning of optical coherent states is proposed and experimentally demonstrated. Its optical realization is based entirely on simple linear optical elements and homodyne detection. The optimality of the presented scheme is only limited by detection inefficiencies. Experimentally we achieved a cloning fidelity of about 65%, which almost touches the optimal value of 2/3.
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).
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 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.
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.
We consider the problem of phase-covariant cloning for coherent states. We show that an experimental scheme based on ideal phase measurement and feedforward outperforms the semiclassical procedure of ideal phase measurement and preparation in terms of fidelity. A realistic scheme where the ideal phase measurement is replaced with double-homodyne detection is shown to be unable to overcome the semiclassical cloning strategy. On the other hand, such a realistic scheme is better than semiclassical cloning based on double-homodyne phase measurement and preparation.
We propose a scheme for continuous-variable quantum cloning of coherent states with phase-conjugate input modes using linear optics. The quantum cloning machine yields $M$ identical optimal clones from $N$ replicas of a coherent state and $N$ its replicas of phase conjugate. This scheme can be straightforwardly implemented with the setup accessible at present since its optical implementation only employs simple linear optical elements and homodyne detection. Compared with the original scheme for continuous variables quantum cloning with phase-conjugate input modes proposed by Cerf and Iblisdir…
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 notions of qubits and coherent states correspond to different physical systems and are described by specific formalisms. Qubits are associated with a two-dimensional Hilbert space and can be illustrated on the Bloch sphere. In contrast, the underlying Hilbert space of coherent states is infinite-dimensional and the states are typically represented in phase space. For the particular case of binary coherent state alphabets these otherwise distinct formalisms can equally be applied. We capitalize this formal connection to analyse the properties of optimally cloned binary coherent states. Seve…
We present a probabilistic cloning scheme operating independently of any phase reference. The scheme is based solely on a phase-randomized displacement and photon counting, omitting the need for non-classical resources and non-linear materials. In an experimental implementation, we employ the scheme to clone coherent states from a phase covariant alphabet and demonstrate that the cloner is capable of outperforming the hitherto best-performing deterministic scheme. An analysis of the covariances between the output states shows that uncorrelated clones can be approached asymptotically. An intrig…
We show that in the case of unknown {\em harmonic oscillator coherent states} it is possible to achieve what we call {\it perfect information cloning}. By this we mean that it is still possible to make arbitrary number of copies of a state which has {\it exactly} the same information content as the original unknown coherent state. By making use of this {\it perfect information cloning} it would be possible to estimate the original state through measurements and make arbitrary number of copies of the estimator. We define the notion of a {\em Measurement Fidelity}. We show that this information …
A scheme for the optimal Gaussian cloning of coherent light states at the light-atoms interface is proposed. The distinct feature of this proposal is that the clones are stored in an atomic quantum memory, which is important for applications in quantum communication. The atomic quantum cloning machine requires only a single passage of the light pulse through the atomic ensembles followed by the measurement of a light quadrature and an appropriate feedback, which renders the protocol experimentally feasible. An alternative protocol, where one of the clones is carried by the outgoing light pulse…
The fidelity of a quantum transformation is strongly linked with the prior partial information of the state to be transformed. We illustrate this interesting point by proposing and demonstrating the superior cloning of coherent states with prior partial information. More specifically, we propose two simple transformations that under the Gaussian assumption optimally clone symmetric Gaussian distributions of coherent states as well as coherent states with known phases. Furthermore, we implement for the first time near-optimal state-dependent cloning schemes relying on simple linear optics and f…
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.
Light fields can be amplified by measuring the field amplitude reflected at a beam splitter of reflectivity R and adding a coherent amplitude proportional to the measurement result to the transmitted field. By applying the quantum optical realization of this amplification scheme to single photon inputs, it is possible to clone the polarization states of photons. We show that optimal cloning of single photon polarization is possible when the gain factor of the amplification is equal to the inverse squareroot of 1-R.
Coherence and entanglement are the two most crucial resources for various quantum information processing tasks. Here, we study the interplay of coherence and entanglement under the action of different three qubit quantum cloning operations. Considering certain well-known quantum cloning machines (input state independent and dependent), we provide examples of coherent and incoherent operations performed by them. We show that both the output entanglement and coherence could vanish under incoherent cloning operations. Coherent cloning operations on the other hand, could be used to construct a uni…
We analyze a reversibility of optimal Gaussian $1\to 2$ quantum cloning of a coherent state using only local operations on the clones and classical communication between them and propose a feasible experimental test of this feature. Performing Bell-type homodyne measurement on one clone and anti-clone, an arbitrary unknown input state (not only a coherent state) can be restored in the other clone by applying appropriate local unitary displacement operation. We generalize this concept to a partial LOCC reversal of the cloning and we show that this procedure converts the symmetric cloner to an a…
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.
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 consider the optimal cloning of quantum coherent states with single-clone and joint fidelity as figures of merit. Both optimal fidelities are attained for phase space translation covariant cloners. Remarkably, the joint fidelity is maximized by a Gaussian cloner, whereas the single-clone fidelity can be enhanced by non-Gaussian operations: a symmetric non-Gaussian 1-to-2 cloner can achieve a single-clone fidelity of approximately 0.6826, perceivably higher than the optimal fidelity of 2/3 in a Gaussian setting. This optimal cloner can be realized by means of an optical parametric amplifier …
It has been shown theoretically that a light amplifier working on the physical principle of stimulated emission should achieve optimal quantum cloning of the polarization state of light. We demonstrate close-to-optimal universal quantum cloning of polarization in a standard fiber amplifier for telecom wavelengths. For cloning 1 --> 2 we find a fidelity of 0.82, the optimal value being 5/6 = 0.83.
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…
A linear optical probabilistic scheme for the optimal cloning of a pair of orthogonally-polarized photons is devised, based on single- and two-photon interferences. It consists in a partial symmetrization device, realized with a modified unbalanced Mach-Zehnder interferometer, followed by two independent Hong-Ou-Mandel interferometers. This scheme has the advantage that it enables quantum cloning without the need for stimulated amplification in a nonlinear medium. It can also be modified so to make an optical two-qubit partial SWAP gate, thereby providing a potentially useful tool to linear op…
We report the experimental demonstration of continuous variable cloning of phase conjugate coherent states as proposed by Cerf and Iblisdir (Phys. Rev. Lett. 87, 247903 (2001)). In contrast to the proposal of Cerf and Iblisdir, the cloning transformation is accomplished using only linear optical components, homodyne detection and feedforward. Three clones are succesfully produced with fidelities about 89%.
We investigate the performances of a selective cloning machine based on linear optical elements and Gaussian measurements, which allows to clone at will one of the two incoming input states. This machine is a complete generalization of a 1 to 2 cloning scheme demonstrated by U. L. Andersen et al. [Phys. Rev. Lett. vol. 94, 240503 (2005)]. The input-output fidelity is studied for generic Gaussian input state and the effect of non-unit quantum efficiency is also taken into account. We show that if the states to be cloned are squeezed states with known squeezing parameter, then the fidelity can b…
We analyze in details a scheme for cloning of Gaussian states based on linear optical components and homodyne detection recently demonstrated by U. L. Andersen et al. [PRL 94 240503 (2005)]. The input-output fidelity is evaluated for a generic (pure or mixed) Gaussian state taking into account the effect of non-unit quantum efficiency and unbalanced mode-mixing. In addition, since in most quantum information protocols the covariance matrix of the set of input states is not perfectly known, we evaluate the average cloning fidelity for classes of Gaussian states with the degree of squeezing and …
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.
We investigate the probabilistic cloning and purification of quantum states. The performance of these probabilistic operations is quantified by the average fidelity between the ideal and actual output states. We provide a simple formula for the maximal achievable average fidelity and we explictly show how to construct a probabilistic operation that achieves this fidelity. We illustrate our method on several examples such as the phase covariant cloning of qubits, cloning of coherent states, and purification of qubits transmitted via depolarizing channel and amplitude damping channel. Our exampl…
A simultaneous, contextual experimental demonstration of the two processes of cloning an input qubit and of flipping it into the orthogonal qubit is reported. The adopted experimental apparatus, a Quantum-Injected Optical Parametric Amplifier (QIOPA) is transformed simultaneously into a Universal Optimal Quantum Cloning Machine (UOQCM) and into a Universal NOT quantum-information gate. The two processes, indeed forbidden in their exact form for fundamental quantum limitations, will be found to be universal and optimal, i.e. the measured fidelity of both processes F<1 will be found close to the…
Quantum coherence (QCh) is considered to be a key ingredient in quantum resource theories and also plays a pivotal role in the design and implementation of various information processing tasks. Consequently, it becomes important for us to create more number coherent quantum states between two labs from a given coherent state shared initially. We call this process as \textbf{Broadcasting of Quantum Coherence (QCh)}. In this work we extensively study the problem of broadcasting of QCh. In order to have a holistic view of broadcasting, we start with a most general representation two qubit mixed s…
The cloning of quantum variables with continuous spectra is investigated. We define a Gaussian 1-to-2 cloning machine, which copies equally well two conjugate variables such as position and momentum or the two quadrature components of a light mode. The resulting cloning fidelity for coherent states, namely $F=2/3$, is shown to be optimal. An asymmetric version of this Gaussian cloner is then used to assess the security of a continuous-variable quantum key distribution scheme that allows two remote parties to share a Gaussian key. The information versus disturbance tradeoff underlying this cont…
We study state-dependent quantum cloning which can outperform universal cloning. 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 expectation value of Pauli's Z operator. Our results can be applied for cloning of an arbitrary mirror-symmetric distribution of qubits on Bloch sphere including in special cases the universal cloning and phase-covariant cloning. We show …
We present the first experimental implementation of a multifunctional device for the optimal cloning of one to two qubits. Previous implementations have always been designed to optimize the cloning procedure with respect to one single type of a priori information about the cloned state. In contrast, our "all-in-one" implementation is optimal for several prominent regimes such as universal cloning, phase-covariant cloning, and also the first ever realized mirror phase-covariant cloning, when the square of the expected value of Pauli's Z operator is known in advance. In all these regimes the exp…
We report on two optical realizations of the $1 \to 2$ asymmetric phase-covariant cloning machines for polarization states of single photons. The experimental setups combine two-photon interference and tunable polarization filtering that enables us to control the asymmetry of the cloners. The first scheme involves a special unbalanced bulk beam splitter exhibiting different splitting ratios for vertical and horizontal polarizations, respectively. The second implemented scheme consists of a balanced fiber coupler where photon bunching occurs, followed by a free-space part with polarization filt…
The quantum cloner machine maps an unknown arbitrary input qubit into two optimal clones and one optimal flipped qubit. By combining linear and non-linear optical methods we experimentally implement a scheme that, after the cloning transformation, restores the original input qubit in one of the output channels, by using local measurements, classical communication and feedforward. This significant teleportation-like method demonstrates how the information is preserved during the cloning process. The realization of the reversion process is expected to find useful applications in the field of mod…
We address the problem of finding optimal CPTP (completely positive, trace preserving) maps between a set of binary pure states and another set of binary generic mixed state in a two dimensional space. The necessary and sufficient conditions for the existence of such CPTP maps can be discussed within a simple geometrical picture. We exploit this analysis to show the existence of an optimal quantum repeater which is superior to the known repeating strategies for a set of coherent states sent through a lossy quantum channel. We also show that the geometrical formulation of the CPTP mapping condi…
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 propose entangled (M+1)-mode quantum states as a multiuser quantum channel for continuous-variable communication. Arbitrary quantum states can be sent via this channel simultaneously to M remote and separated locations with equal minimum excess noise in each output mode. For a set of coherent-state inputs, the channel realizes optimum symmetric 1-to-M cloning at a distance (``telecloning''). It also provides the optimal cloning of coherent states without the need of amplifying the state of interest. The generation of the multiuser quantum channel requires no more than two 10\log_{10}[(\sqrt…
We investigate the multiphoton states generated by high-gain optical parametric amplification of a single injected photon, polarization encoded as a ''qubit''. The experiment configuration exploits the optimal phase-covariant cloning in the high gain regime. The interference fringe pattern showing the non local transfer of coherence between the injected qubit and the mesoscopic amplified output field involving up to 4000 photons has been investigated. A probabilistic new method to extract full information about the multiparticle output wavefunction has been implemented.
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.
We demonstrate that a phase-independent quantum amplifier of a polarization qubit is a complementary amplifier of the heralded qubit amplifier [N. Gisin, S. Pironio and N. Sangouard, Phys. Rev. Lett. 105, 070501 (2010)]. It employs the multi-functional cloner in 1 to 2 copying regime, capable of providing approximate copies of qubits given by various probability distributions, and is optimized for distributions with axial symmetry. Direct applications of the proposed solution are possible in quantum technologies, doubling the range where quantum information is coherently broadcast. It also out…
We report on experimental implementation of the optimal universal asymmetric 1->2 quantum cloning machine for qubits encoded into polarization states of single photons. Our linear optical machine performs asymmetric cloning by partially symmetrizing the input polarization state of signal photon and a blank copy idler photon prepared in a maximally mixed state. We show that the employed method of measurement of mean clone fidelities exhibits strong resilience to imperfect calibration of the relative efficiencies of single-photon detectors used in the experiment. Reliable characterization of the…
We propose an interesting scheme for distributed orbital state quantum cloning with atomic ensembles based on the quantum Zeno dynamics. These atomic ensembles which consist of identical three-level atoms are trapped in distant cavities connected by a single-mode integrated optical star coupler. These qubits can be manipulated through appropriate modulation of the coupling constants between atomic ensemble and classical field, and the cavity decay can be largely suppressed as the number of atoms in the ensemble qubits increases. The fidelity of each cloned qubit can be obtained with analytic r…
We discuss the recently introduced concept of non-deterministic noiseless linear amplification, demonstrating that such an operation can only be performed perfectly with vanishing probability of success. We show that a weak measurement, which imprints the weak value of an observable of a pre-selected and post-selected system onto a probe system, can be used to approximate probabilistic noiseless amplification. This result may be applied to various tasks in continuous variable quantum information, including entanglement concentration, probabilistic cloning, and in quantum repeaters. We discuss …
We present the results of a linear optics photonic implementation of a quantum circuit that simulates a phase covariant cloner, by using two different degrees of freedom of a single photon. We experimentally simulate the action of two mirrored $1\rightarrow 2$ cloners, each of them biasing the cloned states into opposite regions of the Bloch sphere. We show that by applying a random sequence of these two cloners, an eavesdropper can mitigate the amount of noise added to the original input state and therefore prepare clones with no bias but with the same individual fidelity, masking its presenc…
We experimentally demonstrate phase-insensitive linear optical amplification which preserves the idler at the output. Since our amplification operation is unitary up to small excess noise, it is reversible beyond the classical limit. The entanglement between the two output modes is the resource for the reversibility. The amplification gain of 2.0 is demonstrated. In addition, combining this amplifier with a beamsplitter, we also demonstrate approximate cloning of coherent states where an anticlone is present. We investigate the reversibility by reconstructing the initial state from the output …
We pose the question whether the asymptotic equivalence between quantum cloning and quantum state estimation, valid at the single-clone level, still holds when all clones are examined globally. We conjecture that the answer is affirmative and present a large amount of evidence supporting our conjecture, developing techniques to derive optimal asymptotic cloners and proving their equivalence with estimation in virtually all scenarios considered in the literature. Our analysis covers the case of arbitrary finite sets of states, arbitrary families of coherent states, arbitrary phase- and multipha…
One of the most fascinating aspects of quantum mechanics is the principle impossibility of deterministic errorless discrimination of nonorthogonal signals, such as coherent states. On the one hand, it prevents perfect cloning of quantum states and enables secure communication. On the other hand, it makes a grand challenge to reach the ultimate measurement precision. Although the minimum possible error rate (the Helstrom bound) has been known for almost five decades, there is no practical way to achieve it. Developing the realistic optimal measurement strategies to attain the Helstrom bound is …
We investigate the resources needed for secure teleportation of coherent states. We extend continuous variable teleportation to include quantum tele-amplification protocols, that allow non-unity classical gains and a pre-amplification or post-attenuation of the coherent state. We show that, for arbitrary Gaussian protocols and a significant class of Gaussian resources, two-way steering is required to achieve a teleportation fidelity beyond the no-cloning threshold. This provides an operational connection between Gaussian steerability and secure teleportation. We present practical recipes sugge…
We present the experimental realization of optimal symmetric and asymmetric phase-covariant 1->2 cloning of qubit states using fiber optics. State of each qubit is encoded into a single photon which can propagate through two optical fibers. The operation of our device is based on one- and two-photon interference. We have demonstrated creation of two copies of any state of a qubit from the equator of the Bloch sphere. The measured fidelities of both copies are close to the theoretical values and they surpass the theoretical maximum obtainable with the universal cloner.
