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arXiv:quant-ph/0602033 · Journal of Physics B Atomic Molecular and Optical Physics

Continuous variable tripartite entanglement and Einstein–Podolsky–Rosen correlations from triple nonlinearities

M. K. Olsen, Ashton S. Bradley, M. D. Reid

Published May 15, 2006• 25 citations
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Abstract

We compare theoretically the tripartite entanglement available from the use of three concurrent $χ^{(2)}$ nonlinearities and three independent squeezed states mixed on beamsplitters, using the van Loock-Furusawa inequalities. We also define three-mode generalisations of the Einstein-Podolsky-Rosen paradox which are an alternative for demonstrating the inseparability of the density matrix.

DOI
10.1088/0953-4075/39/11/016

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Conference on Lasers and Electro-Optics•Jan 1, 2004•0 citations
We compare theoretically the tripartite entanglement available from the use of three concurrent χ(2)χ^{(2)}χ(2) nonlinearities and three independent squeezed states mixed on beamsplitters, using the van Loock-Furusawa inequalities. We also define three-mode generalisations of the Einstein-Podolsky-Rosen paradox which are an alternative for demonstrating the inseparability of the density matrix.
We develop three-mode generalizations of the Einstein-Podolsky-Rosen paradox, allowing us to define inequalities which may be used to indicate continuous-variable tripartite entanglement. We use these inequalities and an appropriate version of the previously developed van Loock-Furusawa inequalities to theoretically compare the continuous-variable tripartite entanglement available from the use of three concurrent x (2) nonlinearities and from three independent squeezed states mixed on beamsplitters.
We analyze the three-mode correlation properties of the electromagnetic field in a optical parametric oscillator below-threshold. We employ a perturbative expansion of the It\^o equations derived from the positive-P representation of the density matrix. Using the generalized Cauchy-Schwarz inequality, we investigate the genuine quantum nature of the triple correlations between the interacting fields, since in this case continuous-variable entanglement is not detected by the van Loock-Furusawa criterion [Phys. Rev. A 67, 052315 (2003)]. Although not being a necessary condition, these triple correlations are sufficient evidence of tripartite entanglement. Of course, our characterization of the quantum correlations is applicable to non-Gaussian states, which we show to be the case of the optical parametric oscillator below-threshold, provided nonlinear quantum fluctuations are properly taken into account.
Following previous work, we distinguish between genuine NNN-partite entanglement and full NNN-partite inseparability. Accordingly, we derive criteria to detect genuine multipartite entanglement using continuous-variable (position and momentum) measurements. Our criteria are similar but different to those based on the van Loock--Furusawa inequalities, which detect full NNN-partite inseparability. We explain how the criteria can be used to detect the genuine NNN-partite entanglement of continuous variable states generated from squeezed and vacuum state inputs, including the continuous-variable Greenberger-Horne-Zeilinger state, with explicit predictions for up to N=9N=9N=9. This makes our work accessible to experiment. For N=3N=3N=3, we also present criteria for tripartite Einstein-Podolsky-Rosen (EPR) steering. These criteria provide a means to demonstrate a genuine three-party EPR paradox, in which any single party is steerable by the remaining two parties.
A generalization of the 1935 Einstein-Podolsky-Rosen (EPR) argument for measurements with continuous variable outcomes is presented to establish criteria for the demonstration of the EPR paradox, for situations where the correlation between spatially separated subsystems is not perfect. Two types of criteria for EPR correlations are determined. The first type are based on measurements of the variances of conditional probability distributions and are necessary to reflect directly the situation of the original EPR paradox. The second weaker set of EPR criteria are based on the proven failure of (Bell-type) local realistic theories which could be consistent with a local quantum description for each subsystem. The relationship with criteria sufficient to prove entanglement is established, to show that any demonstration of EPR correlations will also signify entanglement. It is also shown how a demonstration of entanglement between two spatially separated subsystems, if not able to interpreted as a violation of a Bell-type inequality, may be interpreted as a demonstration of the EPR correlations. In particular it is explained how the experimental observation of two-mode squeezing using spatially separated detectors will signify not only entanglement but EPR correlations defined in a general sense.
Quantum entanglement shared by more than two partites is the essential bases for developing quantum communication networks and quantum computation. The multipartite entanglement of continuous variables can be generated by distributing or combining squeezed states of light using linear optics. We experimentally obtained a tripartite entangled state of bright optical field by means of a nondegenerate optical parametric amplifier (NOPA) and optical beam-splitters. The NOPA is operated in the state of parametric deamplification by locking the relative phase between the pump laser and the injected seed wave to (2n+1)π, so the generated Einstein-Podolsky-Rosen (EPR) beam for continuous variables is an entangled state of light with the quantum correlation of phase quadratures and the quantum anticorrelation of amplitude quadratures [1]. The output EPR beam consists of two frequency-degenerate optical modes with horizontal and vertical polarizations which are distributed to three parties with two linear beam-splitters. We have theoretically and experimentally demonstrated the three parties are in a tripartite entangled state with three-mode “position” (amplitude quadrature) correlation and relative “momentum” (phase quadrature) correlation [2]-[4]. The obtained GHZ-like state is a fully inseparable tripartite continuous-variables state violating the three inequalities used for the criterion of the inseparability [5][6]. The controlled dense coding quantum communication is experimentally accomplished by exploiting the tripartite entanglement. The three entangled modes are sent to a sender (Alice), a receiver (Bob) and a controller (Claire), respectively. The information transmission capacity of the quantum channel between Alice and Bob is controlled by Claire. The channel capacity completed under Claire’s help is always larger than that without her help.