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College Physics

Solving EPR-type entangled states of continuous variable by virtue of normally ordered product method

Nian-Quan Jiang

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

The method getting the specific form of EPR-type entangled state of continuous variable in Fock representation is proposed.The specific form of bipartite EPR entangled state in Fock representation is derived by using the properties of normally ordered product(NOP) and operating the function of boson operators in the form of NOP.Furthermore,the method is applied to getting the specific form of multi-partite EPR entangled state in Fock representation.So,it is a general method.

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The method getting the specific form of EPR-type entangled state of continuous variable in Fock representation is proposed.The specific form of bipartite EPR entangled state in Fock representation is derived by using the properties of normally ordered product(NOP) and operating the function of boson operators in the form of NOP.Furthermore,the method is applied to getting the specific form of multi-partite EPR entangled state in Fock representation.So,it is a general method.
A new kind of four-mode continuous variable coherent-entangled state is proposed in the Fock space by using the technique of integration within an ordered product, which exhibits both the properties of a coherent state and an entangled state, and spans a complete and orthonormal representation. The conjugate state of the four-mode continuous variable coherent-entangled state is derived by using the Fourier transformation. Moreover, a simple experimental protocol of generating a four-mode continuous variable coherent-entangled state is proposed by using beam splitters. As applications of this four-mode continuous variable coherent-entangled state, a four-mode entangled state and a four-mode squeezing-Fresnel operator are constructed.
We consider tripartite entangled states for continuous variable systems of EPR type, which generalize the famous bipartite CV EPR states (eigenvectors of conjugate choices X 1 − X 2 , P 1 + P 2 , of the systems' relative position and total momentum variables). We give the regularized forms of such tripartite EPR states in second-quantized formulation, and derive their Wigner functions. This is directly compared with the established NOPA-like states from quantum optics. Whereas the multipartite entangled states of NOPA type have singular Wigner functions in the limit of large squeezing, r → ∞ , or tanh r → 1 − (approaching the EPR states in the bipartite case), our regularized tripartite EPR states show singular behaviour not only in the approach to the EPR-type region ( s → 1 in our notation), but also for an additional, auxiliary regime of the regulator . While the s → 1 limit pertains to tripartite CV states with singular eigenstates of the relative coordinates and remaining squeezed in the total momentum, the limit yields singular eigenstates of the total momentum, but squeezed in the relative coordinates. Regarded as expectation values of displaced parity measurements, the tripartite Wigner functions provide the ingredients for generalized CHSH inequalities. Violations of the tripartite CHSH bound ( B 3 ⩽ 2) are established, with in the canonical regime ( s → 1 + ), as well as in the auxiliary regime .