The explicit relationship of photon number tomogram and optical tomogram is obtained for two-mode light. The corresponding integral kernel is expressed in terms of Laguerre polynomials. The connection of the Heisenberg uncertainty relationship with photon statistics is elucidated.
The scheme of photon-number tomography is discussed in the framework of star- product quantization. The connection of dual quantization scheme and observables is reviewed. The quantizer and dequantizer operators and kernels of star-product of tomograms in photon-number tomography scheme and its dual one are presented in explicit form. The fidelity and state purity are discussed in photon-number tomographic scheme, and the expressions for fidelity and purity are obtained in the form of integral of the product of two photon-number tomograms with integral kernel which is presented in explicit form. The properties of quantumness are discussed in terms of inequalities on state photon-number tomograms.
Similar in spirit to the preceding work 9 Fan, H-Y and Hu, L-Y. 2009. Opt. Commun., 282: 3734–3736. [Crossref], [Web of Science ®] , [Google Scholar], 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 .