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Chinese Physics Letters

Properties and Application of Intermediate Entangled State Representation

Pang Qian-Jun

Published May 25, 2005• 4 citations
Publisher

Abstract

We introduce the intermediate entangled state |α⟩ and discuss its properties of |α⟩ λ,ν . We obtain the expression of the two-mode Wigner operator in the |α⟩ λ,ν representation. We find that the projection operator |α⟩ λ,ν λ,ν ⟨α| is just the Radon transformation of the two-mode Wigner operator. Using the |α⟩ λ,ν representation, we successfully obtain the eigenvalues and eigenvectors of a bipartite system coupled in a quite complicated way.

DOI
10.1088/0256-307x/22/6/005

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We introduce the intermediate entangled state |α⟩ and discuss its properties of |α⟩ λ,ν . We obtain the expression of the two-mode Wigner operator in the |α⟩ λ,ν representation. We find that the projection operator |α⟩ λ,ν λ,ν ⟨α| is just the Radon transformation of the two-mode Wigner operator. Using the |α⟩ λ,ν representation, we successfully obtain the eigenvalues and eigenvectors of a bipartite system coupled in a quite complicated way.
The coherent-intermediate-entangled state |α, x ⟩ λ,ν is proposed in the two-mode Fock Space, which exhibits both the properties of the coherent and entangled states. The |α, x ⟩ λ,ν makes up a new quantum mechanical representation, and the completeness relation of |α, x ⟩ λ,ν is proved by virtue of the technique of integral within an ordered product of operators. The corresponding squeezing operators are derived. Furthermore, Generalized P-representation is constructed in the coherent-intermediate-entangled state |α, x ⟩ λ,ν and the Schmidt decomposition of |α, x ⟩ λ,ν is investigated.
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.
As a natural and important extension of Fan's paper (Fan Hong-Yi 2010 Chin. Phys. B 19 040305) by employing the formula of operators' Weyl ordering expansion and the bipartite entangled state representation this paper finds a new two-fold complex integration transformation about the Wigner operator Δ (μ, ν) (in its entangled form) in phase space quantum mechanics, and its inverse transformation. In this way, some operator ordering problems regarding to ( a † 1 – a 2 ) and ( a † 1 – a 2 ) can be solved and the contents of phase space quantum mechanics can be enriched, where a i , a † i are bosonic creation and annihilation operators, respectively.