Verifying angular-position entanglement by Hardy's paradox with multisetting high-dimensional systems

Abstract

High-dimensional entangled states are of crucial importance, as they provide higher channels for quantum information. Angular position, as a continuous variable, provides an infinite Hilbert space theoretically. It is usually represented in discrete bases experimentally because of the convenient modulation of the width of the angular aperture. Thus the combination of the angular apertures will conveniently shape the two photon states. And the entanglement of the two photon states can be naturally demonstrated by Hardy's paradox. Here, by testing Hardy's paradox for two-setting high-dimensional angular subspaces with dimension ranging from 2 to 7 and for multisetting three-dimensional subspaces with setting ranging from 3 to 5, we reveal the high-dimensional entanglement in the angular-position degree of freedom. We show that the high-dimensional angular-position entanglement can yield a much sharper contradiction between the quantum mechanics and classical theories. Our work shows that the angular variable can be considered as an alternative discrete base to provide the high-dimensional entangled states. Thus it may be used as a new platform for quantum information.