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Chinese Physics B

New two-mode intermediate momentum-coordinate representation with quantum entanglement and its application

Shi-Min Xu, Xing-Lei Xu, Hongqi Li, Ji‐Suo Wang

Published Jun 1, 2009• 4 citations
Publisher

Abstract

We construct four linear composite operators for a two-particle system and give common eigenvectors of those operators. The technique of integration within an ordered product (IWOP) of operators is employed to prove that those common eigenvectors are complete and orthonormal. Therefore, a new two-mode intermediate momentum-coordinate representation which involves quantum entanglement for a two-particle system is proposed and applied to some two-body dynamic problems. Moreover, the pure-state density matrix |ξ 1 , ξ 2 〉 C,D C,D 〈ξ 1 , ξ 2 | is a Radon transform of Wigner operator.

DOI
10.1088/1674-1056/18/6/004

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We construct four linear composite operators for a two-particle system and give common eigenvectors of those operators. The technique of integration within an ordered product (IWOP) of operators is employed to prove that those common eigenvectors are complete and orthonormal. Therefore, a new two-mode intermediate momentum-coordinate representation which involves quantum entanglement for a two-particle system is proposed and applied to some two-body dynamic problems. Moreover, the pure-state density matrix |ξ 1 , ξ 2 〉 C,D C,D 〈ξ 1 , ξ 2 | is a Radon transform of Wigner operator.
A unitary transformation matrix, n linear-composite coordinate operators and n linear-composite momentum operators are constructed for an n-particle system, and the complete and orthonormal common eigenvectors of the multi-mode linear composite momentum operators are examined by virtue of the technique of integration within an ordered product of operators. The multi-mode linear composite momentum representation is proposed, and its application to a general two-mode forced quantum harmonic oscillator system with kinetic coupling is presented for solving some dynamic problems.
We show that for tomographic approach there exist two mutual conjugate quantum states | p ,σ,τ⟩ and | x ,μ,ν⟩ (named the intermediate coordinate-momentum representation), and the two Radon transforms of the Wigner operator are just the pure-state density matrices | p ⟩ σ,τσ,τ ⟨ p | and | x ⟩ λ,νλ,ν ⟨ x | respectively. As a result, the tomogram of quantum states can be considered as the module-square of the states' wave function in these two representations. Throughout the paper we fully employ the technique of integration within an ordered product of operators. In this way we establish a new convenient formalism of quantum tomogram.