The impossibility of perfectly copying (or cloning) an unknown 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 cryptography, the link between optimal cloning and light amplification via stimulated emission, and the experimental demonstrations of optimal quantum cloning

The power of matrix product states to describe infinite-size translational-invariant critical spin chains is investigated. At criticality, the accuracy with which they describe ground-state properties of a system is limited by the size of the matrices that form the approximation. This limitation is quantified in terms of the scaling of the half-chain entanglement entropy. In the case of the quantum Ising model, we find S ϳ 1 6 log with high precision. This result can be understood as the emergence of an effective finite correlation length ruling all the scaling properties in the system. We produce six extra pieces of evidence for this finite-scaling, namely, the scaling of the correlation length, the scaling of magnetization, the shift of the critical point, the scaling of the entanglement entropy for a finite block of spins, the existence of scaling functions, and the agreement with analogous classical results. All our computations are consistent with a scaling relation of the form ϳ , with = 2 for the Ising model. In the case of the Heisenberg model, we find similar results with the value ϳ 1.37. We also show how finite-scaling allows us to extract critical exponents. These results are obtained using the infinite time evolved block decimation algorithm which works in the thermodynamical limit and are verified to agree with density-matrix renormalization-group results and their classical analog obtained with the corner transfer-matrix renormalization group.

A transformation achieving the optimal symmetric N → 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.PACS numbers: 03.65.Bz, 42.50.Dv, 89.70.+c Quantum systems cannot be cloned exactly [1], but only approximately. Finding the optimal approximate quantum cloning transformation has been a fundamental issue in quantum information theory for the last five years. In quantum cryptography, for instance, this problem happens to be strongly related to the assessment of security [2]. Cloning has been extensively studied to date for discrete quantum variables, such as quantum bits [3][4][5][6][7][8][9] or d-level systems [10][11][12], since quantum information theory was initially developed for these kinds of systems. Recent progress has shown, however, that continuous spectrum systems might be experimentally simpler to manipulate than their discrete counterparts in order to process quantum information (see, e.g., [13,14]).Stimulated by this progress, we investigate in this Letter the possibility of achieving the cloning of continuousvariables quantum information. Commonly, a distinction is made between universal cloning, if the set of input states contains all possible states for a given Hilbert space dimension, and state-dependent cloning, if the input states are restricted to a certain set which does not contain all possible states. For any Hilbert space dimension, the optimal universal cloner [10][11][12] that clones all possible input states equally well can be constructed from a single family of quantum circuits [15]. This universal cloner reduces to a classical probability distributor in the continuous limit. Besides the universal cloner, quantum cloning of continuous-variable systems has been considered first in a state-dependent context.

No abstract

An optimal universal cloning transformation is derived that produces M copies of an unknown qubit from a pair of orthogonal qubits. For M > 6, the corresponding cloning fidelity is higher than that of the optimal copying of a pair of identical qubits. It is shown that this cloning transformation can be implemented probabilistically via parametric down-conversion by feeding the signal and idler modes of a nonlinear crystal with orthogonally polarized photons.In contrast to classical information, quantum information cannot be copied. This so-called no-cloning theorem [1], which is a direct consequence of the linearity of quantum theory, states that it is impossible to prepare several exact copies (or clones) of an unknown quantum state |ψ . Although exact cloning is forbidden, one can design various quantum cloning machines which produce approximate clones. In particular, much attention has been devoted to the optimal universal cloning machines for qubits, which prepare M identical approximate clones out of N replicas of an unknown qubit, and such that the fidelity of the clones is state-independent [2]. Cloning machines for states in a d-dimensional Hilbert space (qudits) were also investigated [3], as well as continuousvariable cloning machines for coherent states [4].In the limit of an infinite number of clones, the optimal cloning reduces to the optimal quantum measurement. In this context, a very interesting observation has been made by Gisin and Popescu [5] who noted that the information about a direction in space is better encoded into two orthogonal qubits than in two identical ones. If we possess a two-qubit state |ψ, ψ ⊥ with ψ|ψ ⊥ = 0, then we can estimate |ψ with a fidelity 789 [5, 6]. This slightly exceeds the fidelity of the optimal measurement on a qubit pair |ψ, ψ , F || = 3/4. A similar situation occurs for continuous quantum variables. Suppose we want to encode a (randomly chosen) position in phase space. A possible strategy would be to prepare a pair of coherent states |α, α , where the real and imaginary parts of the complex number α represent the phase-space coordinates. However, it is actually better to supply a state |α, α * from which α can be inferred via optimal measurement with a lower error [7]. It can also be shown that the state |α, α * gives an advantage when cloning coherent states: M identical approximate clones of a coherent state |α can be prepared with a higher fidelity from the state |α, α * than from |α, α [8].Motivated by this result, we were led to ask whether a similar situation might also occur for qubits. Can M clones of qubit |ψ be produced from an orthogonal qubit pair |ψ, ψ ⊥ with a higher fidelity than from an identical pair |ψ, ψ ? In this Letter, we answer this question by an affirmative. We present a universal cloning machine acting on an orthogonal qubit pair that approximately implements the transformation |ψ |ψ ⊥ → |ψ ⊗M with the optimal fidelity. Then, we show that this cloning transformation can be implemented probabilistically in quantum optics by use of par...