14 pages LaTex. To appear in the Journal of Modern Optics for the special issue on "Quantum Information: Theory, Experiment and Perspectives". Proceedings of the ESF Conference, Gdansk, July 10-18, 2001
We investigate several classes of state-dependent quantum cloners for three-level systems. These cloners optimally duplicate some of the four maximally-conjugate bases with an equal fidelity, thereby extending the phase-covariant qubit cloner to qutrits. Three distinct classes of qutrit cloners can be distinguished, depending on two, three, or four maximally-conjugate bases are cloned as well (the latter case simply corresponds to the universal qutrit cloner). 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 show that the distribution of information at the output of the quantum cloner can be efficiently controlled via preparation of the quantum cloner. We present a universal cloning network with the help of which asymmetric cloning can be performed.
In this work, we extensively study the problem of broadcasting of entanglement. In the first part of the work, we reconceptualize the idea of state dependent quantum cloning machine, and in that process we introduce different types of state dependent cloners like static and dynamic state dependent cloners. We derive the conditions under which we can make these cloners independent of the input state. In the broadcasting part, as our resource initial state, we start with general two qubit state and consider specific examples like, non maximally entangled state (NME), Werner like state (WS), and …
We consider the quantum cloning of continuous variable entangled states. This is achieved by introducing two symmetric entanglement cloning machines (or e-cloners): a local e-cloner and a global e-cloner; where we look at the preservation of entanglement in the clones under the condition that the fidelity of the clones is maximized. These cloning machines are implemented using simple linear optical elements such as beam splitters and homodyne detection along with squeeze gates. We show that the global e-cloner out-performs the local e-cloner both in terms of the fidelity of the cloned states a…
Cloning machines, that is, transformations that achieve the best approximate copying of a quantum state compatible with the no-cloning theorem, have been a fundamental research topic over the last five years. This study is of particular significance given the close connection between quantum cloning and quantum cryptography: using an optimal cloner generally makes it possible to obtain a tight bound on the best individual eavesdropping strategy in a quantum cryptosystem. In this Chapter, the issue of cloning a continuous-variable quantum system will be analyzed, and a Gaussian cloning transfor…
We study the role of interference in the process of quantum cloning. We show that in order to achieve better than classical cloning of a qubit no interference is needed. In particular, a large class of symmetric universal 1$\to$ 2 qubit cloners exists which achieve the optimal average fidelity 5/6 for such machines, without using any interference. We also obtain optimal average fidelities for interference--free cloning in asymmetric situations, and discuss the relation of the quantum cloners found to the Bužek--Hillery quantum cloner.
Due to the no-cloning theorem, the unknown quantum state can only be cloned approximately or exactly with some probability. There are two types of cloners: universal and state-dependent cloner. The optimal universal cloner has been found and could be viewed as a special state-dependent quantum cloner which has no information about the states. In this paper, we investigate the state-dependent cloning when the state-set contains more than two states. We get some bounds of the global fidelity for these processes. This method is not dependent on the number of the states contained in the state-set.…
We derive a tight upper bound for the fidelity of a universal N to M qubit cloner, valid for any M \geq N, where the output of the cloner is required to be supported on the symmetric subspace. Our proof is based on the concatenation of two cloners and the connection between quantum cloning and quantum state estimation. We generalise the operation of a quantum cloner to mixed and/or entangled input qubits described by a density matrix supported on the symmetric subspace of the constituent qubits. We also extend the validity of optimal state estimation methods to inputs of this kind.
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 establish the best possible approximation to a perfect quantum cloning machine which produces two clones out of a single input. We analyze both universal and state-dependent cloners. The maximal fidelity of cloning is shown to be 5/6 for universal cloners. It can be achieved either by a special unitary evolution or by a novel teleportation scheme. We construct the optimal state-dependent cloners operating on any prescribed two non-orthogonal states, discuss their fidelities and the use of auxiliary physical resources in the process of cloning. The optimal universal cloners permit us to deri…
We show that for any Hilbert-space dimension, the optimal universal quantum cloner can be constructed from essentially the same quantum circuit, i.e., we find a universal design for universal cloners. In the case of infinite dimensions (which includes continuous variable quantum systems) the universal cloner reduces to an essentially classical device. More generally, we construct a universal quantum circuit for distributing qudits in any dimension which acts covariantly under generalized displacements and momentum kicks. The behavior of this covariant distributor is controlled by its initial s…
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 establish a deep connection between the 3 qubit one-to-two phase-covariant quantum cloning network of Fuchs et al. [C. Fuchs, N. Gisin, R.B. Griffiths, C.S. Niu, and A. Peres, Phys. Rev. A 56 no 4, 1163 (1997)], and its economic 2 qubit counterpart due to Niu and Griffiths [Phys.Rev. A 60 no 4, 2764 (1999)]. A general, necessary and sufficient criterion is derived in order to characterize the reducibility of 3 qubit cloners to 2 qubit cloners. When this criterion is fulfilled, economic cloning is possible. We show that the optimal isotropic or universal 3 qubit cloning machine…
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 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…
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…
A generalized universal quantum cloning machine is proposed which allows the input to be arbitrary states in symmetric subspace. And it reduces to the universal quantum cloning machine (UQCM) if the input are identical pure states. The generalized cloner is optimal in the sense we compare the input and output reduced density operators at a single qubit. The result for qubits is extended to arbitrary-dimensional states.
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 …
We investigate the connection between the optimal collective eavesdropping attack and the optimal cloning attack where the eavesdropper employs an optimal cloner to attack the quantum key distribution (QKD) protocol. The analysis is done in the context of the security proof in [Devetak and Winter, Proc. of the Roy. Soc. of London Series A, 461, 207 (2005); Kraus, Gisin and Renner, Phys. Rev. Lett. 95, 080501 (2005)] for discrete variable protocols in d-dimensional Hilbert spaces. We consider a scenario in which the protocols and cloners are equipped with symmetries. These symmetries are used t…
Perfect cloning of a known set of states with arbitrary prior probabilities is possible if we allow the cloner to sometimes fail completely. In the optimal case the probability of failure is at its minimum allowed by the laws of quantum mechanics. Here we show that it is possible to lower the failure rate below that of the perfect probabilistic cloner but the price to pay is that the clones are not perfect; the global fidelity is less than one. We determine the optimal fidelity of a cloner with a Fixed Failure Rate (FFR cloner) in the case of a pair of known states. Optimality is shown to be a…
In this work, we extensively study the problem of broadcasting of entanglement as state dependent versus state independent cloners. We start by re-conceptualizing the idea of state dependent quantum cloning machine (SD-QCM), and in that process, we introduce different types of SD-QCMs, namely, orthogonal and non-orthogonal cloners. We derive the conditions for which the fidelity of these cloners will become independent of the input state. We note that such a construction allows us to maximize the cloning fidelity at the cost of having partial information of the input state. In the discussion o…
We show that optimal universal cloning of the polarization state of photons can be achieved via stimulated emission in three-level systems, both of the Lambda and the V type. We establish the equivalence of our systems with coupled harmonic oscillators, which permits us to analyze the structure of the cloning transformations realized. These transformations are shown to be equivalent to the optimal cloning transformations for qubits discovered by Buzek and Hillery, and Gisin and Massar. The down-conversion cloner discovered previously by some of the authors is obtained as a limiting case. We de…
We derive the transformation for the optimal universal quantum anti-cloner which produces two anti-parallel outputs for a single input state. The fidelity is shown to be 2/3 which is same as the measurement fidelity. We consider a probabilistic quantum anti-cloner and show quantum states can be anti-cloned exactly with non-zero probability and its efficiency is higher than the efficiency of distinguishing between the two states.
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 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 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…
The quantum state space $\cal S$ over a $d$-dimensional Hilbert space is represented as a convex subset of a $D-1$-dimensional sphere $S_{D-1}\subset {\bf{R}}^D$, where $D=d^2-1.$ Quantum tranformations (CP-maps) are then associated with the affine transformations of ${\bf{R}}^D,$ and $N\mapsto M$ {\it cloners} induce polynomial mappings. In this geometrical setting it is shown that an optimal cloner can be chosen covariant and induces a map between reduced density matrices given by a simple contraction of the associated $D$-dimensional Bloch vectors.
We report on experimental implementation of a machine-learned quantum gate driven by a classical control. The gate learns optimal phase-covariant cloning in a reinforcement learning scenario having fidelity of the clones as reward. In our experiment, the gate learns to achieve nearly optimal cloning fidelity allowed for this particular class of states. This makes it a proof of present-day feasibility and practical applicability of the hybrid machine learning approach combining quantum information processing with classical control. Moreover, our experiment can be directly generalized to larger …
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…
In this work, we exhaustively investigate $1 \rightarrow 2$ local and nonlocal broadcasting of entanglement as well as correlations beyond entanglement (geometric discord) using asymmetric Pauli cloners with most general two qubit state as the resource. We exemplify asymmetric broadcasting of entanglement using Maximally Entangled Mixed States. We demonstrate the variation of broadcasting range with the amount of entanglement present in the resource state as well as with the asymmetry in the cloner. We show that it is impossible to optimally broadcast geometric discord with the help of these a…
In this paper we present an approach to quantum cloning with unmodulated spin networks. The cloner is realized by a proper design of the network and a choice of the coupling between the qubits. We show that in the case of phase covariant cloner the XY coupling gives the best results. In the 1->2 cloning we find that the value for the fidelity of the optimal cloner is achieved, and values comparable to the optimal ones in the general N->M case can be attained. If a suitable set of network symmetries are satisfied, the output fidelity of the clones does not depend on the specific choice of the g…
We pose the question whether the asymptotic equivalence between quantum cloning and quantum state estimation, valid at single-copy level, still holds when all the copies are examined jointly. For an N-to-M cloner, we consider the overall fidelity between the state of the M output systems and the state of M ideal copies, and we ask whether the optimal fidelity is attained by a measure-and- prepare protocol in the limit of large M. In order to gain intuition into the general problem, we analyze two concrete examples: i) cloning qubit states on the equator of the Bloch sphere and ii) cloning two-…
We construct a quantum machine which, by using asymmetric cloner, deals with disentangling and broadcasting entanglement in a single unitary evolution. The attainable maximum value of the scaling parameter $s$ for disentangling is identical to that obtained in previous works. The fidelity of the cloning state with respect to the input entangled state is state-dependent.
If black holes were able to clone quantum states, a number of paradoxes in black hole physics would disappear. However, the linearity of quantum mechanics forbids exact cloning of quantum states. Here we show that black holes indeed clone incoming quantum states with a fidelity that depends on the black hole's absorption coefficient, without violating the no-cloning theorem because the clones are only approximate. Perfectly reflecting black holes are optimal universal "quantum cloning machines" and operate on the principle of stimulated emission, exactly as their quantum optical counterparts. …
Universal quantum cloning machines (UQCMs), sometimes called quantum cloners, generate many outputs with identical density matrices, with as close a resemblance to the input state as is allowed by the basic principles of quantum mechanics. Any experimental realization of a quantum cloner has to cope with the effects of decoherence which terminate the coherent evolution demanded by a UQCM. We examine how many clones can be generated within a decoherence time. We compare the time that a quantum cloner implemented with trapped ions requires to produce $M$ copies from $N$ identical pure state inpu…
We, in this paper, analyze the efficacy of an output as a resource from a universal quantum cloning machine in information processing tasks such as teleportation and dense coding. For this, we have considered the $3\otimes 3$ dimensional system (or qutrit system). The output states are found to be NPT states for certain ranges of machine parameters. Using the output state as an entangled resource, we have also studied the optimal fidelities of teleportation and capacities of dense coding protocols with respect to the machine parameters and have made a few interesting observations. Our work is …
Inria, Télécom Paris – LTCI, Institut Polytechnique de Paris（法国国家信息与自动化研究所，巴黎电信学院-信息与通信实验室，巴黎理工学院）；QuSoft & Institute for Logic, Language and Computation & Korteweg-de Vries Institute for Mathematics, University of Amsterdam（QuSoft、逻辑语言与计算研究所、科特维格-德弗里斯数学研究所，阿姆斯特丹大学）；UC Berkeley（加州大学伯克利分校）
In this work, we revisit the problem of finding an admissible region of fidelities obtained after an application of an arbitrary $1 \rightarrow N$ universal quantum cloner which has been recently solved in [A. Kay et al., Quant. Inf. Comput 13, 880 (2013)] from the side of cloning machines. Using group-theory formalism, we show that the allowed region for fidelities can be alternatively expressed in terms of overlaps of pure states with recently found irreducible representations of the commutant $U \otimes U \otimes \ldots \otimes U \otimes U^*$, which gives the characterization of the allowed…
To decide whether a quantum channel is degradable is relatively easy: one has to find at least one example of a degrading quantum channel. But in general, no conclusive criterion exists to show the opposite. Using elementary methods we derive a necessary and sufficient condition to decide under what circumstances the conclusion is unambiguous. The findings lead to an extension of the antidegradability region for qubit and qutrit transpose depolarizing channels. In the qubit case we reproduce the known results for the class of qubit depolarizing channels (due to their equivalence). One of the c…
