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arXiv:quant-ph/0401007 · Physical Review Letters

Identifying Entanglement Using Quantum Ghost Interference and Imaging

Milena D’Angelo, Yoon-Ho Kim, S. P. Kulik, Yanhua H. Shih

Published Jun 8, 2004• 201 citations
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Abstract

We report a quantum interference and imaging experiment which allows identifying the entanglement in momentum and position variables of a two-photon system. The measurements show indeed that the uncertainties in the sum of momenta and in the difference of positions of the entangled two-photon satisfy both EPR inequalities $\ensuremath{\Delta}({k}_{1}+{k}_{2})<\mathrm{min}(\ensuremath{\Delta}{k}_{1},\ensuremath{\Delta}{k}_{2})$ and $\ensuremath{\Delta}({x}_{1}\ensuremath{-}{x}_{2})<\mathrm{min}(\ensuremath{\Delta}{x}_{1},\ensuremath{\Delta}{x}_{2})$. These two inequalities, together, represent a nonclassicality condition. Our measurements provide a direct way to distinguish between quantum entanglement and classical correlation in continuous variables for systems of pairs of photons.

DOI
10.1103/physrevlett.92.233601

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We report a quantum interference and imaging experiment which allows identifying the entanglement in momentum and position variables of a two-photon system. The measurements show indeed that the uncertainties in the sum of momenta and in the difference of positions of the entangled two-photon satisfy both EPR inequalities \ensuremathΔ(k1+k2)<min(\ensuremathΔk1,\ensuremathΔk2)\ensuremath{\Delta}({k}_{1}+{k}_{2})<\mathrm{min}(\ensuremath{\Delta}{k}_{1},\ensuremath{\Delta}{k}_{2})\ensuremathΔ(k1​+k2​)<min(\ensuremathΔk1​,\ensuremathΔk2​) and \ensuremathΔ(x1\ensuremath−x2)<min(\ensuremathΔx1,\ensuremathΔx2)\ensuremath{\Delta}({x}_{1}\ensuremath{-}{x}_{2})<\mathrm{min}(\ensuremath{\Delta}{x}_{1},\ensuremath{\Delta}{x}_{2})\ensuremathΔ(x1​\ensuremath−x2​)<min(\ensuremathΔx1​,\ensuremathΔx2​). These two inequalities, together, represent a nonclassicality condition. Our measurements provide a direct way to distinguish between quantum entanglement and classical correlation in continuous variables for systems of pairs of photons.
Following Einstein, Podolsky, and Rosen (EPR) original argument, we derive a pair of inequalities for the uncertainties in the sum of the momenta and in the difference of the positions of an entangled two-particle system. We prove that only entangled systems can satisfy both EPR-inequalities, in violation of the limits imposed by classical statistics. We propose an operational approach that allows applying the EPR inequalities to two-photon systems, as well. In particular, we demonstrate a scheme that allows implementing the EPR gedanken-experiment and verifying the EPR inequalities on systems of pairs of photons. We report the experimental results obtained in this kind of quantum interference and imaging experiment: SPDC two-photon system satisfies both EPR inequalities. The experimental verification of the EPR inequalities represents a practical way to experimentally distinguish entanglement from classical correlation in momentum and/or position variables for systems of two particles/photons. We emphasize the practical consequences of the EPR inequalities: only entanglement allows one to go beyond the limitations imposed by Heisenberg uncertainty on systems of classically correlated particles/photons. In this context we review a recent experiment of two-photon diffraction and quantum lithography.
Entanglement, a key resource of emerging quantum technologies, describes correlations between particles that defy classical physics. It has been studied extensively on various platforms, but has remained elusive in electron microscopy. Transmission electron microscopes are well-established tools for materials characterisation with unparalleled spatial resolution. They provide control over the preparation and detection of high energy electrons, with largely unexploited potential in the study of many-body quantum correlations. Here, we demonstrate entanglement in electron-photon pairs generated via cathodoluminescence in a transmission electron microscope. Employing coincidence imaging techniques adapted from photonic quantum optics, we reconstruct both near- and far-field ``ghost'' images of periodic transmission masks. By measuring spatial and momentum correlations, we show a violation of the classical uncertainty bound: Δx−2Δk+2=0.502±0.047<1Δx_-^2 Δk_+^2 = 0.502 \pm 0.047<1Δx−2​Δk+2​=0.502±0.047<1. Hence, we demonstrate entanglement in position and momentum -- the continuous variables at the base of most imaging methods, bridging the fields of electron microscopy and quantum optics. Our work paves the way for exploring quantum correlations in free-electron systems and their application to quantum-enhanced imaging techniques on the nanoscale.
One of the most surprising consequences of quantum mechanics is the entanglement of two or more distant particles. In an entangled EPR two-particle system, the value of the momentum (position) for neither single subsystem is determined. However, if one of the subsystems is measured to have a certain momentum (position), the other subsystem is determined to have a unique corresponding value, despite the distance between them. This peculiar behavior of an entangled quantum system has surprisingly been observed experimentally in two-photon temporal and spatial correlation measurements, such as ghost interference and ghost imaging. This article addresses the fundamental concerns behind these experimental observations and to explore the nonclassical nature of two-photon superposition by emphasizing the physics of 2 is not 1 + 1.