Optimal Cloning of Coherent States with a Linear Amplifier and Beam Splitters

Abstract

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.

Cloning of Gaussian states by linear optics

“…Optimal Gaussian cloning can be realized using a phase insensitive amplifier and a beam splitter [10,11]. However, it has been recently shown, theoretically and experimentally, that the parametric amplifier can be replaced by a simpler scheme involving only linear optical components, homodyne detection and a feed-forward loop [12].…”

“…Note that in this Section we are addressing the case of an unknown squeezing parameter ξ (randomly distributed according to a given probability density): when it is known, the optimal strategy in the Gaussian regime is to perform the unsqueezing operation S −1 (ξ) just before the cloning machine, proceed as in the case of coherent states, and, at the output stage, apply the squeezing operation S(ξ) to both the clones which yields a fidelity of 2/3 (independent on the amount of fixed squeezing) as in the coherent state case [10]. However in the case of an unknown squeezing parameter the squeezing action S(ξ) is not known.…”

“…Originally, such a machine was proposed for cloning of qubits and has later been demonstrated experimentally [6]. Shortly after this development, a continuous variable (CV) [7] analog of the qubit quantum cloner was proposed [8,9] and recently it was shown that a CV optimal Gaussian cloner of coherent states can be implemented using an appropriate combination of beam splitters and a single phase insensitive parametric amplifier [10,11]. Although this proposal sounds experimentally promising, the implementation of an efficient phase insensitive amplifier operating at the fundamental limit is a challenging task.…”

Non-Gaussianity of quantum states: An experimental test on single-photon-added coherent states

“…While Gaussian states, defined as states with a Gaussian Wigner function, are known to provide useful resources for tasks such as teleportation [2,3], cloning [4][5][6], or dense coding [7][8][9], there is an ongoing effort to study which protocols are allowed by non-Gaussian resources. The most notable example is certainly their use for an optical quantum computer [10,11], alongside their employment for improving teleportation [12][13][14], cloning [15], and storage [16].…”

Probabilistic implementation of Hadamard and unitary gates

“…We also derive the best transformation efficiencies. [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22] which operate either in a deterministic or probabilistic way. Corresponding to the quantum no-cloning theorem, Pati and Braunstein [23] demonstrated that the linearity of quantum mechanics also forbids one to delete one unknown state ideally against a copy [23], which is called the quantum no-deleting principle.…”