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arXiv:2110.06781 · Physical Review Research

Quantifying tripartite spatial and energy-time entanglement in nonlinear optics

James Schneeloch, Richard J. Birrittella, Christopher C. Tison, Gregory A. Howland, Michael L. Fanto, Paul M. Alsing

Published May 26, 2022• 4 citations
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Abstract

In this work, we provide a means to quantify genuine tripartite entanglement in arbitrary (pure and mixed) continuous-variable states as measured by the tripartite entanglement of formation---a resource-based measure quantifying genuine multipartite entanglement in units of elementary Greenberger-Horne-Zeilinger (GHZ) states called gebits. Furthermore, we predict its effectiveness in quantifying the tripartite spatial and energy-time entanglement in photon triplets generated in cascaded spontaneous parametric down-conversion (SPDC), and find that ordinary nonlinear optics can be a substantial resource of tripartite entanglement.

DOI
10.1103/physrevresearch.4.023153

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•Dec 6, 2013•0 citations
In this work, we provide a means to quantify genuine tripartite entanglement in arbitrary (pure and mixed) continuous-variable states as measured by the tripartite entanglement of formation---a resource-based measure quantifying genuine multipartite entanglement in units of elementary Greenberger-Horne-Zeilinger (GHZ) states called gebits. Furthermore, we predict its effectiveness in quantifying the tripartite spatial and energy-time entanglement in photon triplets generated in cascaded spontaneous parametric down-conversion (SPDC), and find that ordinary nonlinear optics can be a substantial resource of tripartite entanglement.
We lay down a general scheme to quantify the amount of genuine tripartite entanglement present in the spatial and energy-time degrees of freedom of entangled photon triplets using a resource-based measure known as the tripartite entanglement of formation. Quantifying genuine tripartite entanglement relative to a number of maximally entangled three-qubit Greenberger-Horne-Zeilinger (GHZ) states called gebits, the tripartite entanglement of formation serves as a basis of comparison between different tripartite entangled states of differing dimension. Demonstrating genuine tripartite entanglement is doubly challenging because it is not enough to show that each party is inseparable from the other two (which is sufficient only for pure states). Instead, one must rule out all mixtures of all combinations of biseparable states from describing the tripartite state. To meet this challenge, we use entropic measures of the statistics of tripartite systems to simultaneously bound the correlations each party has with the other two, and in so doing place a lower limit to the tripartite entanglement of formation. Even though our measure never over-estimates the entanglement present, we estimate the effectiveness of our technique by determining the exact tripartite entanglement of a triple-gaussian triphoton wavefunction with the same correlations as seen in photon triplets generated in third-order spontaneous parametric down-conversion (SPDC). Between these two efforts, we show that a substantial amount of tripartite entanglement exists in both the spatial and energy-time degrees of freedom of these systems, and propose an experiment that can measure them.
From an operational criterion of physical reality, a quantifier of realism-based nonlocality was recently introduced for two-part quantum states. This measure has been shown to capture aspects that are rather different from Bell nonlocality. Here we take a step further and introduce a tripartite realism-based nonlocality quantifier. We show that this measure reduces to genuine tripartite entanglement for a certain class of pure tripartite states and manifests itself in correlated mixed states even in the absence of quantum correlations. A case study for noisy Greenberger-Horne-Zeilinger (GHZ) and WWW states points out the existence of scenarios where the realism-based nonlocality is monogamous.
We study the properties of bisqueezed tripartite Gaussian states created by two spontaneous parametric down-conversion processes that share a common idler. We give a complete description of the quantum correlations across all partitions, as well as of the genuine multipartite entanglement, obtaining analytical expressions for most of the quantities of interest. We find that the state contains genuine tripartite entanglement, in addition to the bipartite entanglement among the modes that are directly squeezed. We also investigate the effect of homodyne detection of the photons in the common idler mode, and analyze the final reduced state of the remaining two signal modes. We find that this measurement leads to a conversion of the coherence of the two signal modes into entanglement, a phenomenon that can be regarded as a redistribution of quantum resources between the modes. The applications of these results to quantum optics and circuit quantum electrodynamics platforms are also discussed.
High quality entangled photon sources are a key requirement for many promising quantum optical technologies. However, the production of multi-photon entangled states with good fidelity is challenging. Current sources of multi-photon entanglement require the use of post-selection, which limits their usefulness for some applications. It has been an open challenge to create a source capable of directly producing three-photon entanglement. An important step in this direction was achieved with the demonstration of photon triplets produced by a new process called cascaded downconversion, but these previous measurements were not sufficient to show whether these photons were in an entangled state and only had detection rates of five triplets per hour. In this thesis, we show the first demonstration of a direct source of three-photon entanglement. Our source is based on cascaded downconversion, and we verify that it produces genuine tripartite entanglement in two degrees of freedom: energy-time and polarization. \n \nThe energy-time entanglement is similar to a three-particle generalization of an Einstein-Podolski-Rosen state; the three photons are created simultaneously, yet the sum of their energies is well defined, which is an indication of energy-time entanglement. To prove it, we use time-bandwidth inequalities which check for genuine tripartite entanglement. Our measurements show that the state violates the inequalities with what constitute, to the best of our knowledge, the strongest violation of time-bandwidth inequalities in a tripartite continuous-variable system to date. \n \nWe create polarization entanglement by modifying our experimental setup so that two downconversion processes producing orthogonally polarized triplets interfere to create Greenberger-Horne-Zeilinger states. By using highly efficient superconducting nanowire single photon detectors, we improve the detected triplet rate by 2 orders of magnitude to 660 triplets per hour. We characterize the state using quantum state tomography, and find a fidelity of 86\\% with the ideal state, beating the previous best value for a three-photon entangled state fidelity measured by tomography. We also use the state to perform two tests of local realism. We violate the Mermin and Svetlichny inequalities by 10 and 5 standard deviations respectively, the latter being the strongest violation to date. Finally, we show that, unlike previous sources of tree-photon entanglement, our source can be used as a source of heralded Bell pairs. We demonstrate this by measuring a CHSH inequality with the heralded Bell pairs, and by reconstructing their state using quantum state tomography.