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

Abstract

Similar in spirit to the preceding work 9, where the relation between optical Fresnel transformation and quantum tomography is revealed, we explore this kind of relationship in a two-mode entangled case. We show that under the two-mode Fresnel transformation the entangled state density becomes density operator , which is just the Radon transform of the two-mode Wigner operator in entangled form, i.e.

where F ₂ is a two-mode Fresnel operator in quantum optics, and s , r are the complex-value expressions of (A,B,C,D). So the probability distribution is the tomography (Radon transform of the two-mode Wigner function), and gives a tomogram. Similarly, we find

where is the conjugated state to .

Keywords:

• optical Fresnel transformation
• quantum tomography
• two-mode entangled state

Acknowledgement

This work was supported by the National Natural Science Foundation of China (Grant Nos. 10874174 and 11047133) and the President Foundation of Chinese Academy of Science, a grant from the Key Programs Foundation of Ministry of Education of China (No. 210115) and the Research Foundation of the Education Department of Jiangxi Province of China (No. GJJ10097).