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Communications in Theoretical Physics

Entangled Eigenvectors of Scalar Fields ϕ 1 ( x ) - ϕ 2 ( x ) and Π 1 ( x ) + Π 2 ( x )

Fan Hong‐Yi, Chen ZengBing

Published Dec 15, 2000• 0 citations
Publisher

Abstract

We first demonstrate the precise Einstein-Podolsky-Rosen entanglement, which is inherent in quantum mechanics, in the context of quantum theory of complex scalar fields. We explicitly derive the entangled common eigenstates of ϕ 1 ( x ) - ϕ 2 ( x ) and Π 1 ( x ) + Π 2 ( x ) ( ϕ 1 and ϕ 2 are two real components of the complex scalar field, and Π 1 and Π 2 are the corresponding conjugate fields) in the Fock space. The entangled eigenstates span a complete and orthonormal representation.

DOI
10.1088/0253-6102/34/4/741

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We first demonstrate the precise Einstein-Podolsky-Rosen entanglement, which is inherent in quantum mechanics, in the context of quantum theory of complex scalar fields. We explicitly derive the entangled common eigenstates of ϕ 1 ( x ) - ϕ 2 ( x ) and Π 1 ( x ) + Π 2 ( x ) ( ϕ 1 and ϕ 2 are two real components of the complex scalar field, and Π 1 and Π 2 are the corresponding conjugate fields) in the Fock space. The entangled eigenstates span a complete and orthonormal representation.
For complex scalar fields ( x ) and ( x ), we introduce the entangled states which are their common eigenvectors. The states possessing complete and orthonormal properties are a field-theoretical generalization of the Einstein-Podolsky-Rosen pair states in quantum mechanics. Based on the new representation, we obtain the complete and orthonormal states { q }, r , the common eigenvectors of the charge operator and ( x ) ( x ) , with which the charge lowering and raising operators can be naturally defined. The applications of our formulation are briefly discussed.
We demonstrate the Einstein-Podolsky-Rosen (EPR) correlations in the Gaussian trial wavefunctional widely used in the Schrödinger representation of the self-interacting complex scalar fields (CSF). This demonstration is based on the explicit eigenstates ||ξ⟩, which possess the precise EPR entanglement and constitute a complete and orthonormal representation, of CSF ϕ and ϕ † in the Fock space. To show the entanglement of the Gaussian trial wavefunctional, we evaluate in the ⟨ξ||-representation its `entangled' Wigner functional resembling the Wigner function of the two-mode squeezed vacuum state, whose quantum entanglement is well known.
Our primary purpose of this work is to explicitly construct the general multipartite Einstein–Podolsky–Rosen (EPR) entangled state in multi-mode Fock space for a system with different masses of particles, which makes up a new quantum mechanical representation owing to completeness relation and orthogonal property. Its entanglement can be seen more clearly by analyzing its standard Schmidt decomposition. In addition, some applications of the multipartite entanglement are proposed including deriving the generalized Wigner operator and squeezing operator.
Based on Bohr’s ideas two families of states ωθ and ωϕ sharing the same perfect correlation of the Einstein-Podolsky-Rosen state and maximally violating Bell’s inequalities are constructed. Unlike their finite-dimensional counterpartner, these states are not unitarily equivalent. Hence they cannot be transformed into each other by physical operations of local algebras. This is a new entanglement phenomenon emerging in infinite-dimensional systems. Due to the uncertainty principle the existence of unbounded observables depends on the states ωθ and ωϕ. Therefore the manipulation of unbounded observables in quantum information processes cannot provide information for all states. Especially, the canonical unbounded observables q̂i and p̂i, i=1,2 of individual particles do not exist in the GNS representations associated with ωθ and ωϕ, and hence properties of individual particles cannot be obtained in such states.