# Wave-Particle duality in Single-Photon Entanglement

## TL;DR

This paper introduces a novel method to verify single-photon entanglement using Bell's inequality in wave space, highlighting the wave-particle duality and offering new quantum information schemes.

## Contribution

It proposes a new wave-space Bell inequality verification method and defines a wave state measurement, advancing understanding of single-photon entanglement.

## Key findings

- Strong violation of Bell inequality confirms single-photon entanglement.
- Wave state measurement provides additional information in quantum information processing.
- Fourier transform relates wave and photon-number representations, illustrating wave-particle duality.

## Abstract

The simplest single-photon entanglement is the entanglement of the vacuum state and the single-photon state between two path modes. The verification of the existence of single-photon entanglement has attracted extensive research interests. Here, based on the construction of Bell's inequality in wave space, we propose a new method to verify single photon entanglement. Meanwhile, we define the wave state in two-dimensional space relative to the photon number state, and propose a method to measure it. Strong violation of Bell inequality based on joint measurement of wave states indicates the existence of single photon entanglement with certainty. Wave state entanglement obtained from Fourier transform of single photon entanglement and the corresponding measurement protocols will provide us with more information-carrying schemes in the field of quantum information. The difference in the representation in photon-number space and wave space implies the wave-particle duality of single photon entanglement.