No abstract

In the framework of thermo-field dynamics, invented by Umezawa et al., we construct a mixed coherent state representation of density operator ρ. This new representation is useful because it provides an approach to retrieve ρ from its c-number solution of master equations in the entangled state representation.

We study optical Fresnel transforms by finding the appropriate quantum mechanical SU(1,1) squeezing operators which are composed of quadratic combination of canonical operators. In one-mode case, the squeezing operator's matrix element in the coordinate basis is just the kernel of one-dimensional generalized Fresnel transform (GFT); while in two-mode case, the matrix element of the squeezing operator in the entangled state basis leads to the two-dimensional GFT kernel. The work links optical transforms in wave optics to generalized squeezing transforms in quantum optics.KEY WORDS: Fresnel transform; SU(1,1) squeezing operator; the entangled state.

Based on the conception of quantum entanglement of Einstein-Podolsky-Rosen we construct generalized phase space representation associated with the entangled state |Γ e , which is endowed with definite physical meaning. The set of states make up a complete and non-orthogonal representation. The Weyl ordered form of |Γ ee Γ| is derived which clearly exhibit the statistical behavior of marginal distribution of |Γ ee Γ| . The minimum uncertainty relation obeyed by |Γ e is also demonstrated.