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 evidence up to d=7. An economical phase-covariant cloner can nevertheless be constructed for d>2, albeit with a lower fidelity than that of the optimal cloner requiring an ancilla. Finally, using again an ansatz on cloning machines, we show that an economical version of the Fourier-covariant cloner, which optimally clones the computational basis and its Fourier transform, is also possible only in dimension d=2.
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 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 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 propose a scheme to enhance the fidelity of symmetric quantum cloning machine using a weak measurement. By adjusting the intensity of weak measurement parameter $p$, we obtain the copies with different optimal fidelity. Choosing proper value of $p$, we can obtain the perfect copies for initial qubits. In this paper, we focus on $1-2$ quantum cloning for two nonorthogonal states. Sets containing more than two linear independent states are also discussed briefly. Due to weak measurements being probabilistic, we obtain high fidelity at risk of probability. If the weak measurement successes, we…
While the no-cloning theorem, which forbids the perfect copying of quantum states, is well-known as one of the defining features of quantum mechanics, the question of how well the theory allows a state to be cloned is yet to be completely solved. In this paper, rigorous solutions to the problem of M to N asymmetric cloning of qudits are obtained in a number of interesting cases. The central result is the solution to the 1 to N universal asymmetric qudit cloning problem for which the exact trade-off in the fidelities of the clones for every N and d is derived. Analogous results are proven for q…
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-…
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…
We consider a sequential implementation of the optimal quantum cloning machine of Gisin and Massar and propose optimization protocols for experimental realization of such a quantum cloner subject to the real-life restrictions. We demonstrate how exploiting the matrix-product state (MPS) formalism and the ensuing variational optimization techniques reveals the intriguing algebraic structure of the Gisin-Massar output of the cloning procedure and brings about significant improvements to the optimality of the sequential cloning prescription of Delgado et al [Phys. Rev. Lett. 98, 150502 (2007)]. O…
Attempts at cloning a quantum system result in the introduction of imperfections in the state of the copies. This is a consequence of the no-cloning theorem, which is a fundamental law of quantum physics and the backbone of security for quantum communications. Although such perfect copies are prohibited, a quantum state may be copied with maximal accuracy via various optimal cloning schemes. Optimal quantum cloning, which lies at the border of the physical limit imposed by the no-signalling theorem and the Heisenberg uncertainty principle, has been experimentally realized for low dimensional p…
We investigate the universal asymmetric cloning of states in a Hilbert space of arbitrary dimension. We derive the class of optimal and fully asymmetric 1->3 cloners, which produce three copies, each having a different fidelity. A simple parametric expression for the maximum achievable cloning fidelity triplets is then provided. As a side-product, we also prove the optimality of the 1->2 asymmetric cloning machines that have been proposed in the literature.
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 a new class of quantum cloning machines that equally duplicate all real states in a Hilbert space of arbitrary dimension. By using the no-signaling condition, namely that cloning cannot make superluminal communication possible, we derive an upper bound on the fidelity of this class of quantum cloning machines. Then, for each dimension d, we construct an optimal symmetric cloner whose fidelity saturates this bound. Similar calculations can also be performed in order to recover the fidelity of the optimal universal cloner in d dimensions.
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.
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…
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 optimal cloning transformation for two pairs of orthogonal states of two-dimensional quantum systems, and derive the corresponding optimal fidelities.
We have found a quantum cloning machine that optimally duplicates the entanglement of a pair of $d$-dimensional quantum systems. It maximizes the entanglement of formation contained in the two copies of any maximally-entangled input state, while preserving the separability of unentangled input states. Moreover, it cannot increase the entanglement of formation of all isotropic states. For large $d$, the entanglement of formation of each clone tends to one half the entanglement of the input state, which corresponds to a classical behavior. Finally, we investigate a local entanglement cloner, whi…
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…
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.
The quantum mechanical no-cloning theorem for pure states is generalized and transfered to the quantum logics with a conditional probability calculus in a rather abstract, though simple and basic fashion without relying on a tensor product construction or finite dimension as required in other generalizations.
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…
We analyze quantum algorithms for cloning of a quantum measurement. Our aim is to mimic two uses of a device performing an unknown von Neumann measurement with a single use of the device. When the unknown device has to be used before the bipartite state to be measured is available we talk about 1 -> 2 learning of the measurement, otherwise the task is called 1 -> 2 cloning of a measurement. We perform the optimization for both learning and cloning for arbitrary dimension of the Hilbert space. For 1 -> 2 cloning we also propose a simple quantum network that realizes the optimal strategy.
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…
Not all unitary operations upon a set of qubits can be implemented by sequential interactions between each qubit and an ancillary system. We analyze the specific case of sequential quantum cloning 1->M and prove that the minimal dimension D of the ancilla grows linearly with the number of clones M. In particular, we obtain D = 2M for symmetric universal quantum cloning and D = M+1 for symmetric phase-covariant cloning. Furthermore, we provide a recipe for the required ancilla-qubit interactions in each step of the sequential procedure for both cases.
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…
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…
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 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 analyze various eavesdropping strategies on a quantum cryptographic channel. We present the optimal strategy for an eavesdropper restricted to a two-dimensional probe, interacting on-line with each transmitted signal. The link between safety of the transmission and the violation of Bell's inequality is discussed. We also use a quantum copying machine for eavesdropping and for broadcasting quantum information.
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 review our recent work on the universal (i.e. input state independent) optimal quantum copying (cloning) of qubits. We present unitary transformations which describe the optimal cloning of a qubit and we present the corresponding quantum logical network. We also present network for an optimal quantum copying ``machine'' (transformation) which produces N+1 identical copies from the original qubit. Here again the quality (fidelity) of the copies does not depend on the state of the original and is only a function of the number of copies, N. In addition, we present the machine which universaly …
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…
I use the recently proposed framework of ontological models [Harrigan et al., arXiv:0709.1149v2] to obtain economical models for results of tomographically complete sets of measurements on finite-dimensional quantum systems. I describe a procedure that simplifies the models by decreasing the number of necessary ontic states, and present an explicit model with just 33 ontic states for a qutrit.
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 …
Encrypted Cloning, Absolute Maximal Entanglement and Quantum Secret Sharing
National University of Singapore（新加坡国立大学）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）；Department of Physics, National University of Singapore（新加坡国立大学物理系）；Quantum Bridge Technologies Inc.（量子桥技术有限公司）；University of Toronto（多伦多大学）；Department of Electrical and Computer Engineering, University of Toronto（多伦多大学电气与计算机工程系）
We investigate the role of symmetric quantum cloning machines (QCMs) in quantifying the mutual incompatibility of quantum observables. Specifically, we identify a cloning-based incompatibility measure whereby the incompatibility of a set of observables maybe quantified in terms of how well a uniform ensemble of their eigenstates can be cloned via a symmetric QCM. We show that this new incompatibility measure $\mathcal{Q}_{c}$ is {\it faithful} since it vanishes only for commuting observables. We prove an upper bound for $\mathcal{Q}_{c}$ for any set of observables in a finite-dimensional syste…
We prove generic versions of the no-cloning and no-broadcasting theorems, applicable to essentially {\em any} non-classical finite-dimensional probabilistic model that satisfies a no-signaling criterion. This includes quantum theory as well as models supporting ``super-quantum'' correlations that violate the Bell inequalities to a larger extent than quantum theory. The proof of our no-broadcasting theorem is significantly more natural and more self-contained than others we have seen: we show that a set of states is broadcastable if, and only if, it is contained in a simplex whose vertices are …
We study a (k,m)-threshold controlling scheme for controlled quantum teleportation. A standard polynomial coding over GF(p) with prime p > m-1 needs to distribute a d-dimensional qudit with d >= p to each controller for this purpose. We propose a scheme using m qubits (two-dimensional qudits) for the controllers' portion, following a discussion on the benefit of a quantum control in comparison to a classical control of a quantum teleportation.
Optimal quantum cloning is the process of making one or more copies of an arbitrary unknown input quantum state with the highest possible fidelity. All reported demonstrations of quantum cloning have so far been limited to copying two-dimensional quantum states, or qubits. We report the experimental realization of the optimal quantum cloning of four-dimensional quantum states, or ququarts, encoded in the polarization and orbital angular momentum degrees of freedom of photons. Our procedure, based on the symmetrization method, is also shown to be generally applicable to quantum states of arbitr…
In this study, we consider a quantum version of multicast network coding as a multicast protocol for sending universal quantum clones (UQCs) from a source node to the target nodes on a quantum network. By extending Owari et al.'s previous results for symmetric UQCs, we derive a protocol for multicasting $1\rightarrow 2$ ($1\rightarrow 3$) {\it asymmetric} UQCs of a $q^r$-dimensional state to two (three) target nodes.Our protocol works under the condition that each edge on a quantum network represented by an undirected graph $G$ transmits a $q$-dimensional state. There exists a classical solvab…
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 introduce a quantum cloning bound which we apply to a straightforward and relatively direct security proof of the prepare-and-measure Bennett-Brassard 1984 (BB84) quantum key distribution (QKD) protocol against collective attacks. The approach we propose is able to handle the practical problem of source and detector alignment imprecisions in a simple way. Specifically, we derive a keyrate bound for a BB84 implementation in which Alice's source emits four given but arbitrary pure states, where the usual equivalence between prepare-and-measure and entanglement-based QKD no longer applies. Our…
We study the mapping which occurs when a single qubit in an arbitrary state interacts with another qubit in a given, fixed state resulting in some unitary transformation on the two qubit system which, in effect, makes two copies of the first qubit. The general problem of the quality of the resulting copies is discussed using a special representation, a generalization of the usual Schmidt decomposition, of an arbitrary two-dimensional subspace of a tensor product of two 2-dimensional Hilbert spaces. We exhibit quantum circuits which can reproduce the results of any two qubit copying machine of …
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…
We consider a generalisation of Ekert's entanglement-based quantum cryptographic protocol where qubits are replaced by qu$N$its (i.e., N-dimensional systems). In order to study its robustness against optimal incoherent attacks, we derive the information gained by a potential eavesdropper during a cloning-based individual attack. In doing so, we generalize Cerf's formalism for cloning machines and establish the form of the most general cloning machine that respects all the symmetries of the problem. We obtain an upper bound on the error rate that guarantees the confidentiality of quNit generali…
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 …
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 establish a connection between optimal quantum cloning and optimal state estimation for d-dimensional quantum systems. In this way we derive an upper limit on the fidelity of state estimation for d-dimensional pure quantum states and, furthermore, for generalized inputs supported on the symmetric subspace.
We show that it is possible to clone quantum states to arbitrary accuracy in the presence of a Deutschian closed timelike curve (D-CTC), with a fidelity converging to one in the limit as the dimension of the CTC system becomes large---thus resolving an open conjecture from [Brun et al., Physical Review Letters 102, 210402 (2009)]. This result follows from a D-CTC-assisted scheme for producing perfect clones of a quantum state prepared in a known eigenbasis, and the fact that one can reconstruct an approximation of a quantum state from empirical estimates of the probabilities of an informationa…
We introduce an infinite sequence of quantum channels for which the Holevo capacity is additive. The channel series is closely related to the quantum channels arising from universal quantum cloning machines. The additivity proof is motivated by a special property the studied channels enjoy: the property of conjugate degradability. As a consequence of the announced proof, we also provide an easy way of proving the additivity of the Holevo capacity for the original Unruh channel for which the quantum capacity is already known. Consequently, we present not only an infinite series of finite-dimens…
The orbital angular momentum (OAM) of light, associated with a helical structure of the wavefunction, has a great potential for quantum photonics, as it allows attaching a higher dimensional quantum space to each photon. Hitherto, however, the use of OAM has been hindered by its difficult manipulation. Here, exploiting the recently demonstrated spin-OAM information transfer tools, we report the first observation of the Hong-Ou-Mandel coalescence of two incoming photons having nonzero OAM into the same outgoing mode of a beam-splitter. The coalescence can be switched on and off by varying the i…
In the first part of this Dissertation, I study the differences between LOCC (local operations and classical communication) and the more general class of separable operations. I show that the two classes coincide for the case of pure bipartite state input, and derive a set of important consequences. Using similar techniques I also generalize the no-cloning theorem when restricted to separable operations and show that cloning becomes much more restrictive, by providing necessary (and sometimes sufficient) conditions. In the second part I investigate graph states and graph codes with carrier qud…
