Abstract
Based on the two mutually conjugate tripartite entangled state representations (TESR), we introduce the 3-mode optical entangled fractional Fourier transform (EFrFT) through a new non-unitary bosonic operator realization of the SU(2) generators and the tripartite squeezing operator. The EFrFT, which is characteristic of the eigenmodes being three-variable Hermite polynomials, is not the direct product of three single-variable FrFTs. We show that the EFrFT adapts to the transform between two mutually conjugate quantum mechanical TESR, i.e., we can recast the EFrFT into the matrix element of expiπ2−αJ++J− in the TESR. In addition, we define the two functions' convolution in the EFrFT scheme and obtain the convolution theorem using the TESR. The derivation is concise and rigorous because our calculation is based on Dirac's powerful representation theory.
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