A new entangled state |η;θ〉 is proposed by the technique of integral within an ordered product. A generalized Hadamard transformation is derived by virtue of |η;θ〉, which plays a role of Hadamard transformation for (â 1 sin θ - â 2 cos θ) and (â 1 cos θ + â 2 sin θ).
Based on the newly-developed coherent-entangled state representation we propose the so-called Fresnel–Hadamard complementary transformation, which is unitary. The new unitary operator plays the role of both Fresnel transformation for and Hadamard transformation for respectively. Physically, and could be a symmetric beamsplitter's two output fields. We show that the two transformations are concisely expressed in the coherent-entangled state representation as a projective operator in integration form.