Relation between quantum tomography and optical Fresnel transform
Abstract
Corresponding to optical Fresnel transformation characteristic of ray transfer matrix elements (A;B;C;D); AD-BC = 1, there exists Fresnel operator F(A;B;C;D) in quantum optics, we show that under the Fresnel transformation the pure position density |x><x| becomes the tomographic density |x>rs,rs<x|, which is just the Radon transform of the Wigner operator, i.e., F|x><x|F(+) = |x>rs,rs<x|= ∫ dx'dp'delta[x-(Dx'-Bp')]*Wigner operator where 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 Wigner function), and the tomogram of a state |phi> is just the wave function of its Fresnel transformed state F|phi>, i.e. rs<x||phi>= <x|F(+)|phi>. Similarly, we find F|p><p|F(+) = |p>rs,rs<p|= ∫ dx'dp'delta[x-(Ap'-Cx')]*Wigner operator.