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arXiv:1505.05690

A critical evaluation of bipartite entanglement and steering measures in a genuinely tripartite entangled system

M. K. Olsen

Published May 21, 2015• 0 citations
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Abstract

A large number of inequalities have been proposed for the detection of bipartite continuous variable entanglement and Einstein-Podolsky-Rosen steering. Many of these are based on either measured or inferred variances and are relatively easily measured via homodyne detection of optical outputs. In this work we examine a number of bipartite inequalities in the process of cascaded downconversion and sum-frequency generation inside an optical cavity. This has previously been predicted to be a potential source of continuous-variable tripartite entanglement, therefore we know that there are no separable bipartitions. In this work we investigate the performance of the chosen inequalities in the stable, below threshold regime. We show that detection of the existing entanglement is sensitive to the actual inequalities chosen, with some criteria missing it completely. Our results are explained by the fact that violation of the chosen inequalities is sufficient but not necessary to demonstrate the non-classical relationships between the modes, and highlights the outstanding problem of developing a reliable continuous-variable entanglement measure.

DOI
10.48550/arxiv.1505.05690

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A large number of inequalities have been proposed for the detection of bipartite continuous variable entanglement and Einstein-Podolsky-Rosen steering. Many of these are based on either measured or inferred variances and are relatively easily measured via homodyne detection of optical outputs. In this work we examine a number of bipartite inequalities in the process of cascaded downconversion and sum-frequency generation inside an optical cavity. This has previously been predicted to be a potential source of continuous-variable tripartite entanglement, therefore we know that there are no separable bipartitions. In this work we investigate the performance of the chosen inequalities in the stable, below threshold regime. We show that detection of the existing entanglement is sensitive to the actual inequalities chosen, with some criteria missing it completely. Our results are explained by the fact that violation of the chosen inequalities is sufficient but not necessary to demonstrate the non-classical relationships between the modes, and highlights the outstanding problem of developing a reliable continuous-variable entanglement measure.
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.
Einstein-Podolsky-Rosen (EPR) steering is a form of quantum nonlocality which is intermediate between entanglement and Bell nonlocality. EPR steering is a resource for quantum key distribution that is device independent on only one side in that it certifies bipartite entanglement when one party's device is not characterized while the other party's device is fully characterized. In this work, we introduce two types of genuine tripartite EPR steering, and derive two steering inequalities to detect them. In a semi-device-independent scenario where only the dimensions of two parties are assumed, the correlations which violate one of these inequalities also certify genuine tripartite entanglement. It is known that Alice can demonstrate bipartite EPR steering to Bob if and only if her measurement settings are incompatible. We demonstrate that quantum correlations can also detect tripartite EPR steering from Alice to Bob and Charlie, even if Charlie's measurement settings are compatible.
Einstein-Podolsky-Rosen steering is a kind of powerful nonlocal quantum resource in quantum information processing such as quantum cryptography and quantum communication. Many criteria have been proposed in the past few years to detect steerability, both analytically and numerically, for bipartite quantum systems. We propose effective criteria for tripartite steerability and genuine tripartite steerability of three-qubit quantum states by establishing connections between the tripartite steerability (resp. genuine tripartite steerability) and the tripartite entanglement (resp. genuine tripartite entanglement) of certain corresponding quantum states. From these connections, tripartite steerability and genuine tripartite steerability can be detected without using any steering inequalities. The "complex cost" of determining tripartite steering and genuine tripartite steering can be reduced by detecting the entanglement of the newly constructed states in the experiment. Detailed examples are given to illustrate the power of our criteria in detecting the (genuine) tripartite steerability of tripartite states.
We study multipartite entanglement, the generation of Einstein-Podolsky-Rosen (EPR) states, and quantum steering in a three-mode optomechanical system composed of an atomic ensemble located inside a single-mode cavity with a movable mirror. The cavity mode is driven by a short laser pulse, has a nonlinear parametric-type interaction with the mirror and a linear beam-splitter-type interaction with the atomic ensemble. There is no direct interaction of the mirror with the atomic ensemble. A threshold effect for the dynamics of the system is found, above which the system works as an amplifier and below which as an attenuator of the output fields. The threshold is determined by the ratio of the coupling strengths of the cavity mode to the mirror and to the atomic ensemble. It is shown that above the threshold, the system effectively behaves as a two-mode system in which a perfect bipartite EPR state can be generated, while it is impossible below the threshold. Furthermore, a fully inseparable tripartite entanglement and even further a genuine tripartite entanglement can be produced above and below the threshold. In addition, we consider quantum steering and examine the monogamy relations that quantify the amount of bipartite steering that can be shared between different modes. It is found that the mirror is more capable for steering of entanglement than the cavity mode. The two-way steering is found between the mirror and the atomic ensemble despite the fact that they are not directly coupled to each other, while it is impossible between the output of cavity mode and the ensemble which are directly coupled to each other.
Einstein-Podolsky-Rosen (EPR) steering has potential applications in quantum information processing. Here the genuine tripartite EPR steering is investigated in cascaded nonlinear process of spontaneous parametric down-conversion cascaded with a sum-frequency generation in an optical cavity for a wide range of the nonlinear parameters. The threshold properties of the cascaded nonlinear precess are also analyzed both below and without threshold regime, respectively. The genuine tripartite EPR steering is demonstrated based on the criteria for genuine multipartite EPR steering. Our scheme of the generation of genuine tripartite EPR steering can be used as a suggestion for the potential experiments and the applications in quantum communication and computation.
Two Kerr-squeezed optical beams can be combined in a beam splitter to produce non-Gaussian continuous-variable entangled states. We characterize the non-Gaussian nature of the output by calculating the third-order cumulant of quadrature variables and predict the level of entanglement that could be generated by evaluating the Duan-Simon and Reid Einstein-Podolsky-Rosen criteria. These states have the advantage over Gaussian states and non-Gaussian measurement schemes in that the well known, efficient, and proven technology of homodyne detection may be used for their characterization. A physical demonstration maintaining the important features of the model could be realized using optical fibers, beam splitters, and homodyne detection.