# Relation between Optical Fresnel transformation and quantum tomography in two-mode entangled case

## TL;DR

This paper explores the relationship between two-mode optical Fresnel transformations and quantum tomography, showing how entangled states relate to Radon transforms of the two-mode Wigner function, thus linking optical transformations with quantum state reconstruction.

## Contribution

It extends previous work by analyzing the two-mode entangled case, revealing how Fresnel transformations correspond to Radon transforms of the two-mode Wigner operator in quantum optics.

## Key findings

- Fresnel transformation maps entangled states to Radon-transformed Wigner functions.
- Probability distributions of Fresnel quadrature phase relate directly to quantum tomography.
- Similar results are obtained in the frequency domain.

## Abstract

Similar in spirit to the preceding work [Opt. Commun. 282 (2009) 3734] where the relation between optical Fresnel transformation and quantum tomography is revealed, we study this kind of relationship in the two-mode entangled case. We show that under the two-mode Fresnel transformation the bipartite entangled state density |eta><eta| becomes density operator F_2|eta><eta|F_2 ^{dag}=|eta>_{r,s}<eta|, which is just the Radon transform of the two-mode Wigner operator (sigma,gama) in entangled form, where F_2 is an two-mode Fresnel operator in quantum optics, and s,r are the complex-value expression of (A, B, C,D). So the probability distribution for the Fresnel quadrature phase is the {tomography (Radon transform of the two-mode Wigner function), correspondingly, {s,r}_<eta|phi>=<eta|F_2{dag}|phi>. Similarly, we find a simial conclusion in the `frequency` domain.