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Acta Sinica Quantum Optica

New Entangled State Representation and Its Property

Jian-ming Du

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

In this paper,we give the explicit form of the common eigenvectors |η〉 of 2-1 and 1-2 by the use of the integration within an ordered product(IWOP) technique,and the Schmidt decomposition of |η〉 is also given.In addition,we derive another entangled state |ξ〉 which is the conjugate state of |η〉,the inner product 〈ξ|η〉 is also calculated.Finally,the new two-mode squeezing operator S2=μ∫d2η π|μη〉〈η| is constructed and the squeezed property is analyzed simultaneously.

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In this paper,we give the explicit form of the common eigenvectors |η〉 of 2-1 and 1-2 by the use of the integration within an ordered product(IWOP) technique,and the Schmidt decomposition of |η〉 is also given.In addition,we derive another entangled state |ξ〉 which is the conjugate state of |η〉,the inner product 〈ξ|η〉 is also calculated.Finally,the new two-mode squeezing operator S2=μ∫d2η π|μη〉〈η| is constructed and the squeezed property is analyzed simultaneously.
Since the commutation relation [x1-x2,a1+a2]=0,their common eigenvector |β,x〉 is constructed by using the technique of integration within an ordered product(IWOP) of operators.This state exhibits the properties of the coherent and entangled states.Its entangled properties and the relations with the EPR entangled state can be confirmed by Schmidt decomposition and the unilateral integral for|β,x〉.The superposition of the states |β=β1+iβ2,x〉 along β1 axis yields the ideal two-mode entangled state.
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.
Using the technique of integration within an ordered product of operators we construct a generalized two-mode entangled state, which can be generated by an asymmetrical beam splitter (BS). Some important properties of this state, such as orthogonality and Schmidt decomposition, are also discussed by deriving the expression of BS operator in coordinate representation. As its applications, to conjugate state, obtain operator identities, generate new squeezing operators (squeezed state) are also presented. It is shown that the fidelity of quantum teleportation can be enhanced under certain case by using the asymmetrical new squeezed state as entangled resource.
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.
We introduce the quantum Hadamard operator in continuum state vector space and find that it can be decomposed into a single-mode squeezing operator and a position-momentum mutual transform operator. The two-mode Hadamard operator in bipartite entangled state representation is also introduced, which involves the two-mode squeezing operator and |η⟩ ↔ ξ⟩ mutual transformation operator, where |η⟩ and ξ⟩ are mutual conjugate entangled states. All the discussions are proceeded by virtue of the IWOP technique.
We show that the best way to study the relationship between quantum entanglement and quantum squeezing for the multimode case is through constructing the new n -pair entangled state representation . The explicit form of the 2 n -mode squeezing operator which squeezes the n -pair entangled state can be directly derived via the transition from to where M is an n × n complex matrix, and the technique of integration within an ordered product of operators. We also analyze the squeezing properties of the 2 n -mode squeezed state for M being Hermitian, and obtain the variances of the 2 n -mode quadrature operators in the 2 n -mode squeezed vacuum state with a concise result; the condition for reaching the minimum of the uncertainty relationship is also investigated.