The Multimode Entangled State with Three States Superposition on the (N+1)-dimensional Hilbert Space and Their N-th Power Difference Squeezing Properties

According to the principle of linear superposition of quantum mechanics, the multimode entangled states vertical bar psi((3)) > 2(q)is formed on the finite-dimensional (N+1) Hilbert space by linearly superposing three quantum states: the multimode complex conjugate coherent state vertical bar{Z(j)*} > multimode complex conjugate imaginary coherent statel vertical bar{Z(j)*} > (2q) and multimode vacuum state vertical bar{O-j} > (2q) and their difference squeezing properties of generalized nonlinear equal-power N-th power is studied by utilizing the general theory of multirnode squeezed states. The results show that on the finite -dimensional Hilbert space,while some conditions are satisfied, the two quadratures of the statel vertical bar psi((3)) > (2q) present the equal-power N-th power difference squeezing properties, but under some other conditions, the difference squeezing effects of two quadratures can be displayed at the same time. The squeezed depth or squeezed degree in this space is different from that on the full-dimensional Hilbert space. The later result is not in conformity with the uncertainty principle. It is called "two -sided difference squeezing.