The multimode entangled state with three states superposition on the (N+1)-dimensional Hilbert space and their N-th power difference squeezing properties

Abstract

According to the principle of linear superposition of quantum mechanics, the multimode entangled states (as stated below) is formed on the finite-dimensional(N+1) Hilbert space by linearly superposing three quantum states: the multimode complex conjugate coherent state (as stated below), multimode complex conjugate imaginary coherent state (as stated below) and multimode vacuum state (as stated below), and their difference squeezing properties of generalized nonlinear equal-power N-th power is studied by utilizing the general theory of multimode squeezed states. The results show that on the finite -dimensional Hilbert space, while some conditions are satisfied, the two quadratures of the multimode entangled states (as stated below) 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.