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 analyze the problem of approximate quantum cloning when the quantum state is between two latitudes on the Bloch's sphere. We present an analytical formula for the optimized 1-to-2 cloning. The formula unifies the universal quantum cloning (UQCM) and the phase covariant quantum cloning.
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 study the phase-covariant quantum cloning machine for qudits, i.e. the input states in d-level quantum system have complex coefficients with arbitrary phase but constant module. A cloning unitary transformation is proposed. After optimizing the fidelity between input state and single qudit reduced density opertor of output state, we obtain the optimal fidelity for 1 to 2 phase-covariant quantum cloning of qudits and the corresponding cloning transformation.
We derive the optimal N to M phase-covariant quantum cloning for equatorial states in dimension d with M=kd+N, k integer. The cloning maps are optimal for both global and single-qudit fidelity. The map is achieved by an ``economical'' cloning machine, which works without ancilla.
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 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.
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…
We consider an N -> M quantum cloning transformation acting on pure two-level states lying on the equator of the Bloch sphere. An upper bound for its fidelity is presented, by establishing a connection between optimal phase covariant cloning and phase estimation. We give the explicit form of a cloning transformation that achieves the bound for the case N=1, M=2, and find a link between this case and optimal eavesdropping in the quantum cryptographic scheme BB84.
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.
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 t…
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.
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 consider cloning transformations of d-dimensional states of the form e^{iϕ_0}|0> + e^{iϕ_1}|1> +...+ e^{iϕ_{d-1}}|d-1> that are covariant with respect to rotations of the phases ϕ_i's. The optimal cloning maps are easily obtained using a well-defined general characterization of state-dependent 1 -> 2 cloning transformations in arbitrary dimensions. Our results apply to symmetric as well as asymmetric cloners, so that the balance between the fidelity of the two clones can be analyzed.
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…
We show that all Macroscopic Quantum Superpositions (MQS) based on phase-covariant quantum cloning are characterized by an anomalous high resilence to the de-coherence processes. The analysis supports the results of recent MQS experiments and leads to conceive a useful conjecture regarding the realization of complex decoherence - free structures for quantum information, such as the quantum computer.
We consider the entanglement manipulation capabilities of the universal covariant quantum cloner or quantum processor circuit for quantum bits. We investigate its use for cloning a member of a bipartite or a genuine tripartite entangled state of quantum bits. We find that for bipartite pure entangled states a nontrivial behavior of concurrence appears, while for GHZ entangled states a possibility of the partial extraction of bipartite entanglement can be achieved.
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…
We propose an effective scheme for realizing a long-range quantum state phase-covariant cloning between two qubits in fiber-nanocavity system via an adiabatic passage. Since no cavity (fiber) photons or excited levels of the nitrogen-vacancy (NV) center are populated during the whole process, the scheme is immune to the decay of cavity (fiber) and spontaneous emission of the NV center. The strictly numerical simulation shows that the fidelity is high even in the presence of realistic imperfections.
We show that the quantum states generated by universal optimal quantum cloning of a single photon represent an universal set of quantum superpositions resilient to decoherence. We adopt Bures distance as a tool to investigate the persistence ofquantum coherence of these quantum states. According to this analysis, the process of universal cloning realizes a class of quantum superpositions that exhibits a covariance property in lossy configuration over the complete set of polarization states in the Bloch sphere.
The influence of the relativistic covariance requirement on the optimality of the symmetric state-dependent 1 -> 2 cloning machine is studied. Namely, given a photonic qubit whose basis is formed from the momentum-helicity eigenstates, the change to the optimal cloning fidelity is calculated taking into account the Lorentz covariance unitarily represented by Wigner's little group. To pinpoint some of the interesting results, we found states for which the optimal fidelity of the cloning process drops to 2/3 which corresponds to the fidelity of the optimal classical cloner. Also, an implication …
The experimental realization of optimal symmetric phase-covariant 1->2 cloning of qubit states is presented. The qubits are represented by polarization states of photons generated by spontaneous parametric down-conversion. The experiment is based on the interference of two photons on a custom-made beam splitter with different splitting ratios for vertical and horizontal polarization components. From the measured data we have estimated the implemented cloning transformation using the maximum-likelihood method. The result shows that the realized transformation is very close to the ideal one and …
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 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.
It is known that in phase covariant quantum cloning the equatorial states on the Bloch sphere can be cloned with a fidelity higher than the optimal bound established for universal quantum cloning. We generalize this concept to include other states on the Bloch sphere with a definite $z$ component of spin. It is shown that once we know the $z$ component, we can always clone a state with a fidelity higher than the universal value and that of equatorial states. We also make a detailed study of the entanglement properties of the output copies and show that the equatorial states are the only states…
Here, asymmetric phase-covariant quantum cloning machines are defined and trade-off between qualities of their outputs and its impact on entanglement properties of the outputs are studies. In addition, optimal families among these cloners are introduced and also their entanglement properties are investigated. An explicit proof of optimality is presented for the case of qubits, which is based on the no-signaling condition. Our optimality proof can also be used to derive an upper bound on trade-off relations for a more general class of optimal cloners which clone states on a specific orbit of th…
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.
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…
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 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…
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…
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 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 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 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 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.
We explore the possibility of achieving optimal joint measurements of noncommuting observables on a single quantum system by performing conventional measurements of commuting self adjoint operators on optimal clones of the original quantum system. We consider the case of both finite dimensional and infinite dimensional Hilbert spaces. In the former we study the joint measurement of three orthogonal components of a spin 1/2, in the latter we consider the case of the joint measurements of any pair of noncommuting quadratures of one mode of the electromagnetic field. We show that universally cova…
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 introduce a quantum cellular automaton that achieves approximate phase-covariant cloning of qubits. The automaton is optimized for 1-to-2N economical cloning. The use of the automaton for cloning allows us to exploit different foliations for improving the performance with given resources.
We address the general problem of removing correlations from quantum states while preserving local quantum information as much as possible. We provide a complete solution in the case of two qubits, by evaluating the minimum amount of noise that is necessary to decorrelate covariant sets of bipartite states. We show that two harmonic oscillators in arbitrary Gaussian state can be decorrelated by a Gaussian covariant map. Finally, for finite-dimensional Hilbert spaces, we prove that states obtained from most cloning channels (e.g., universal and phase-covariant cloning) can be decorrelated only …
Macroscopically populated quantum superpositions pose a question to what extent macroscopic world obeys quantum mechanical laws. Recently such superpositions for light, generated by optimal quantum cloner, were demonstrated. They are of fundamental and technological interest. We present numerical methods useful for modeling of these states. Their properties are governed by Gaussian hypergeometric function, which cannot be reduced to neither elementary nor easily tractable functions. We discuss the method of efficient computation of this function for half integer parameters and moderate value o…
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 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 describe a unified framework of phase covariant multi user quantum transformations for d-dimensional quantum systems. We derive the optimal phase covariant cloning and transposition tranformations for multi phase states. We show that for some particular relations between the input and output number of copies they correspond to economical tranformations, which can be achieved without the need of auxiliary systems. We prove a relation between the optimal phase covariant cloning and transposition maps, and optimal estimation of multiple phases for equatorial states.
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 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…
Performances of the symmetric universal and phase-covariant cloning transformations and entanglement cloners -- qubit case -- are investigated when the initial state of the hardware or the original state to be cloned is weakly coupled to a thermal environment. Different behaviors of each of these transformations are analyzed and contrasted with the ideal cases.
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.
