Multipartite Entanglement for Continuous Variables: A Quantum Teleportation Network

Abstract

We show that one single-mode squeezed state distributed among N parties using linear optics suffices to produce a truly N -partite entangled state for any nonzero squeezing and arbitrarily many parties. From this N -partite entangled state, via quadrature measurements of N − 2 modes, bipartite entanglement between any two of the N parties can be 'distilled', which enables quantum teleportation with an experimentally determinable fidelity better than could be achieved in any classical scheme.PACS numbers: 03.65.Bz, 42.50.Dv Entanglement is seen as an essential ingredient in quantum communication and computation. For example, it enables quantum teleportation which was originally proposed for systems of discrete variables [1]. Later, quantum teleportation was also proposed for continuous variables [2,3]. The simplest teleportation schemes rely on bipartite entanglement, the entanglement of a pair of systems shared by two parties. For pure states, this kind of entanglement is well-understood and can be quantified [4]. Multipartite entanglement, the entanglement shared by more than two parties, is much more difficult to quantify [5]. Yet in the laboratory, the creation of tripartite discrete-variable entanglement, yielding so-called GHZ states [6], has been reported for single-photon polarization states [7] and using nuclear magnetic resonance [8].Continuous-variable quantum teleportation of arbitrary coherent states has been realized experimentally with bipartite entanglement built from two single-mode squeezed vacuum states combined at a beamsplitter [9]. In the absence of entanglement the best mean fidelity of the reconstructed coherent states is F = 1 2 [10]. Experimentally, F = 0.58 ± 0.02 was achieved. Though this limits our attention to the teleportation of a rather modest set of non-orthogonal states, the fidelity gives a clear experimental signal for the presence of entanglement.Now it is known that even one single-mode squeezed state incident on a beamsplitter yields a bipartite entangled state [11]. This result is in agreement with entropic measures of bipartite pure-state entanglement [12]. If one single-mode squeezed state were distributed among N parties using linear optics would we obtain a truly N -partite entangled state? We will show that we can answer this question using the fidelity criterion for teleporting unknown coherent states. In particular, we will see that one single-mode squeezed state is sufficient to allow quantum teleportation between any two of the N parties with the help of all other parties. The assistance by the other N − 2 parties only relies on local measurements and classical communication. Due to these N − 2 measurements, bipartite entangled states are 'distilled' from the initial N -partite entangled state.The 'position' and 'momentum' of a 1-D wavepacket (units-free withh = 1 2 as in Ref.[13]) are the electric quadrature amplitudes representing the quantum state of a single polarization of a single transverse mode of electromagnetic radiation. We define the action of an i...

Entanglement conditions for two-mode states: Applications

“…It has been shown how to produce genuinely entangled multimode states by van Loock and Braunstein [29]. We now want to give some simple conditions for determining whether a three-mode state is genuinely entangled, and to give an example of such a state that is not Gaussian and whose entanglement can be demonstrated by these conditions.…”

Experimental demonstration of quantum teleportation of a squeezed state

“…Teleportation of quantum entanglement, i.e., entanglement swapping has been also realized [9,10]. Furthermore, CV teleportation has been extended to a multipartite protocol known as a quantum teleportation network [11,12].…”

Separability properties of three-mode Gaussian states

“…The study of CV multipartite entanglement was initiated in [23,24], where a scheme was suggested to create pure CV N -party entanglement using squeezed light and N −1 beamsplitters. In fact, this discussion indicates that tripartite entanglement has already been created (though not investigated or detected) in the CV quantum teleportation experiment [14].…”