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arXiv:0806.1784

Fidelity, purity and entanglement of two-mode spatially Gaussian-entangled light fields in Turbulence Atmosphere

Li‐Gang Wang, Shi-Yao Zhu

Published Jun 11, 2008• 0 citations
arXivHTMLPDF

Abstract

In this paper, we investigate the propagation of two-mode spatially Gaussian-entangled quantum light fields passing through the turbulence atmosphere. From the propagation formula of the two-mode wave function in the position representation, we have derived the analytical expressions for the fidelity, purity and logarithmic negativity (entanglement) of the resulting quantum state after the long-distance atmospheric transportation. Based on the derived formulae, the effects of the atmospheric turbulences on the evolutions of quantum properties of the resulting two-mode quantum state are discussed in detail under different input parameters of the initial two-mode quantum state. The results show that the maximal distributing distance of quantum entanglement is strongly dependent on the atmospheric conditions: when the atmospheric turbulence becomes stronger and stronger, the maximal distance becomes shorter and shorter, and both the fidelity and purity decrease quicker and quicker as functions of propagating distances. Under a certain atmospheric condition, with the increasing of the input entanglement of the initial two-mode spatially Gaussian-entangled quantum state, the maximal distributing distance for preserving the entanglement gradually increases and always has a saturated (upper) limitation, and both the evolutions of the fidelity and purity are affected by the input parameters of the initial two-mode quantum state, Finally the optimal parameters of the input two-mode quantum state with the fixed input entanglement are discussed in order to obtain the optimal transfer distribution of the quantum entanglement over a long distance under a certain atmosphere. Our theoretical results are very helpful for building the distribution of the quantum entanglement via free-space atmosphere link.

DOI
10.48550/arxiv.0806.1784

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In this paper, we investigate the propagation of two-mode spatially Gaussian-entangled quantum light fields passing through the turbulence atmosphere. From the propagation formula of the two-mode wave function in the position representation, we have derived the analytical expressions for the fidelity, purity and logarithmic negativity (entanglement) of the resulting quantum state after the long-distance atmospheric transportation. Based on the derived formulae, the effects of the atmospheric turbulences on the evolutions of quantum properties of the resulting two-mode quantum state are discussed in detail under different input parameters of the initial two-mode quantum state. The results show that the maximal distributing distance of quantum entanglement is strongly dependent on the atmospheric conditions: when the atmospheric turbulence becomes stronger and stronger, the maximal distance becomes shorter and shorter, and both the fidelity and purity decrease quicker and quicker as functions of propagating distances. Under a certain atmospheric condition, with the increasing of the input entanglement of the initial two-mode spatially Gaussian-entangled quantum state, the maximal distributing distance for preserving the entanglement gradually increases and always has a saturated (upper) limitation, and both the evolutions of the fidelity and purity are affected by the input parameters of the initial two-mode quantum state, Finally the optimal parameters of the input two-mode quantum state with the fixed input entanglement are discussed in order to obtain the optimal transfer distribution of the quantum entanglement over a long distance under a certain atmosphere. Our theoretical results are very helpful for building the distribution of the quantum entanglement via free-space atmosphere link.
Understanding the influence of atmospheric turbulence on the propagation of entangled biphoton states is essential for free-space quantum communication protocols. Using the extended Huygens-Fresnel principle and the Kolmogorov turbulence model, we derive an analytical expression for the combined density operator of the signal and idler fields generated via SPDC, following propagation through separate turbulent channels. By expressing this density operator in the continuous position basis, we show how the spatial correlations between signal and idler persist through turbulence despite the loss of state purity, as they transition from being quantum to classical in nature. We further identify a finite range of propagation distances over which the angle-OAM entanglement is maximized, which provides valuable insights for designing free-space quantum communication links operating over several kilometers through the turbulent atmosphere.
We provide a rigorous treatment of the entanglement properties of two-mode Gaussian states in atmospheric channels by deriving and analyzing the input-output relations for the corresponding entanglement test. A key feature of such turbulent channels is a nontrivial dependence of the transmitted continuous-variable entanglement on coherent displacements of the quantum state of the input field. Remarkably, this allows one to optimize the entanglement certification by modifying local coherent amplitudes using a finite, but optimal amount of squeezing. In addition, we propose a protocol which, in principle, renders it possible to transfer the Gaussian entanglement through any turbulent channel over arbitrary distances. Therefore, our approach provides the theoretical foundation for advanced applications of Gaussian entanglement in free-space quantum communication.
The entangled quantum state of a photon pair propagating through atmospheric turbulence suffers decay of entanglement due to the scintillation it experiences. In this paper, we investigate the robustness against this decay for different qutrit states. An infinitesimal propagation equation is used to obtain the density matrix as a function of the propagation distance and the tangle is used to quantify the entanglement between a pair of qutrits. We consider the evolution of various initial states as they propagate through turbulence. Using optimization of the parameters that define the initial state, we obtain expressions for bipartite qutrit states that retain their initial entanglement longer than the initially maximally entangled states.
Due to the transfer of the angular spectrum of the pump beam to the two-photon state in spontaneous parametric downconversion, the generated twin photons are entangled in their spatial degrees of freedom. This spatial entanglement can be observed through correlation measurements in any set of modes in which one may choose to perform measurements. Choosing, e.g., a Hermite-Gaussian (HG) set of spatial modes as a basis, one can observe correlations present in their spatial degrees of freedom. In addition, these modes can be used as alphabets for quantum communication. For global quantum communication purposes, we derive an analytic expression for two-photon detection probability in terms of HG modes, taking into account the effects of the turbulent atmosphere. Our result is more general as it accounts for the propagation of both signal and idler photons through the atmosphere, as opposed to other works considering one photon's propagation in vacuum. We show that while the restrictions on both the parity and order of the downconverted HG fields no longer hold, due to the crosstalk between modes when propagating in the atmosphere, the crosstalk is not uniform: there are more robust modes that tend to keep the photons in them. These modes can be employed in order to increase the fidelity of quantum communication.
Free-space optical communication based on orbital angular momentum (OAM) offers advantages such as high security, large information capacity, and high-speed transmission. However, turbulence can induce random perturbations to the wavefront, thereby affecting communication performance. Hence, accurately measuring turbulence intensity is crucial. We delve into the quantum Fisher information (QFI) of the spatial coherence length, a pivotal parameter in atmospheric turbulence. Within the framework of free-space optical channels, we conduct a comprehensive analysis of the QFI in models encompassing both single and biphoton systems. As photons propagate through free space, the QFI regarding turbulence parameter initially increases and then decreases, due to the coupling and decoherence between the photons and turbulence. Increasing the number of OAM modes can significantly enhance the QFI regarding turbulence parameters. We considered situations closer to practical preparation, our results indicate that the impact of entanglement on estimation precision depends on the purity of the prepared entangled state. Moreover, we discuss and compare the variations in the QFI of various parameters upon considering backward scattering (BS). We believe our work is the first theoretical attempt towards the exploration of estimating the turbulence parameter via QFI.
Quantum entanglement is an essential element for building the backbone of quantum information systems. Our particular interest lies in long-range distribution of entangled photons to facilitate secure data transfer in free space. To achieve this, we rely on photon pairs generated in such a way that their polarization characteristics represent the corresponding qubit states and have a high degree of correlation in measurements. The main focus of this paper is integrity of the quantum states in free-space channels. When transmitted in atmosphere, classical signals suffer from wave front distortions caused by the spatial and temporal fields of the refractive index. However, this mechanism does not have the same bearing on qubit values and their correlation. We study the effects of turbulence on quantum states by utilizing a laboratory testbed that includes an atmospheric chamber developed by the AFRL. It uses a system of controlled components capable of creating various turbulence conditions. When polarized signals are passed through the atmospheric chamber, we analyze the corresponding quantum states and evaluate the degree of entanglement using our mathematical models and existing metrics.