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

摘要

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 |η> <η| becomes density operator f_2|η><η|F ?2=|η>s,r s,r<η|, which is just the Radon transform of the two-mode Wigner operator Δ(σ, γ) in entangled form, i.e. |η>s,rs,r<η|= π∫d_2γd_2σδ(η~(2-) Dσ2+Bγ1)δ(η1- Dσ1-Bγ2)Δ(σ,γ) 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 | s,r<η|ψ>|2 is the tomography (Radon transform of the two-mode Wigner function), and s,r<η|ψ>=<η|F?2|ψ> gives a tomogram. Similarly, we find F 2|ξ> <ξ|F?2=|ξ> s,r s,r<ξ|=π∫d2σd 2γδ(ξ1-Aσ1- Cγ2)δ(ξ2- Aσ2+Cγ1)Δ(σ,γ) where |ξ> is the conjugated state to |η>.