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Physica Scripta

The complex fractional Fourier transform, the complex Wigner transform and the entangled Wigner operator in EPR entangled state representation

Nanrun Zhou, Liyun Hu, Hong-Yi Fan

Published Aug 22, 2008• 4 citations
Publisher

Abstract

Based on the Einstein–Podolsky–Rosen (EPR) entangled state representation, we show that the complex Wigner transform for the complex function ψ(η) turns out to be the quantum statistical average of the entangled Wigner operator in the state |ψ⟩. The complex fractional Fourier transform is also introduced, which corresponds to a rotation of the complex Wigner function. Thus, the intrinsic relation between the complex Wigner transform and the EPR entangled state is revealed.

DOI
10.1088/0031-8949/78/03/035003

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Based on the Einstein–Podolsky–Rosen (EPR) entangled state representation, we show that the complex Wigner transform for the complex function ψ(η) turns out to be the quantum statistical average of the entangled Wigner operator in the state |ψ⟩. The complex fractional Fourier transform is also introduced, which corresponds to a rotation of the complex Wigner function. Thus, the intrinsic relation between the complex Wigner transform and the EPR entangled state is revealed.
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.
For a tripartite entangled system, based on the newly constructed tripartite Einstein–Podolsky–Rosen entangled state, we find the entangled state representation of the Wigner operator. The corresponding Wigner function then leads us to obtain the physical marginal distributions. We emphasize that, for an entangled-particle system, the physical meaning of the Wigner distribution function should lie in that its marginal distributions give the probability of finding the particles in an entangled way.
We find a new complex integration-transform which can establish a new relationship between a two-mode operator’s matrix element in the entangled state representation and its Wigner function. This integration keeps modulus invariant and therefore invertible. Based on this and the Weyl–Wigner correspondence theory, we find a two-mode operator which is responsible for complex fractional squeezing transformation. The entangled state representation and the Weyl ordering form of the two-mode Wigner operator are fully used in our derivation which brings convenience.
In a recent paper of Optics Letter (Fan Hongyi and Lu Hailiang, Optics Letters 28 (2003) 680) the complex fractional Fourier transform (CFFRT) is introduced, which is different from the usual two-dimensional real fractional Fourier transforms. In the present work we illuminate its definition from the point of view of rotation of the complex form of Wigner distributions as well as from the view of optical propagation mechanism in graded-index medium. The newly constructed quantum mechanical entangled state representations are used to link CFFRT to the Wigner distributions.