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Chinese Optics Letters

New fractional entangling transform and its quantum mechanical correspondence

Shuang Xu, Liyun Hu, Jie‐Hui Huang

Published Jan 1, 2015• 2 citations
Publisher

Abstract

In this Letter, a new fractional entangling transformation (FrET) is proposed, which is generated in the entangled state representation by a unitary operator expfiθðab † þ a † bÞg where aðbÞ is the Bosonic annihilate operator.The operator is actually an entangled one in quantum optics and differs evidently from the separable operator, expfiθða † a þ b † bÞg, of complex fractional Fourier transformation.The additivity property is proved by employing the entangled state representation and quantum mechanical version of the FrET.As an application, the FrET of a two-mode number state is derived directly by using the quantum version of the FrET, which is related to Hermite polynomials.

DOI
10.3788/col201513.030801

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In this Letter, a new fractional entangling transformation (FrET) is proposed, which is generated in the entangled state representation by a unitary operator expfiθðab † þ a † bÞg where aðbÞ is the Bosonic annihilate operator.The operator is actually an entangled one in quantum optics and differs evidently from the separable operator, expfiθða † a þ b † bÞg, of complex fractional Fourier transformation.The additivity property is proved by employing the entangled state representation and quantum mechanical version of the FrET.As an application, the FrET of a two-mode number state is derived directly by using the quantum version of the FrET, which is related to Hermite polynomials.
In this Letter, a new fractional entangling transformation(Fr ET) is proposed, which is generated in the entangled state representation by a unitary operator expfiθeabt abTg where aebT is the Bosonic annihilate operator. The operator is actually an entangled one in quantum optics and differs evidently from the separable operator, expfiθeaa t bbTg, of complex fractional Fourier transformation. The additivity property is proved by employing the entangled state representation and quantum mechanical version of the Fr ET. As an application, the Fr ET of a two-mode number state is derived directly by using the quantum version of the Fr ET, which is related to Hermite polynomials.
A new type of entangled fractional squeezing transformation (EFrST) has been theoretically proposed for 2D entanglement [ Front. Phys . 10, 100302 (2015)]. In this paper, we shall extend this case to that of 3D entanglement by introducing a type of three-mode entangled state representation, which is not the product of three 1D cases. Using the technique of integration within an ordered product of operators, we derive a compact unitary operator corresponding to the 3D fractional entangling transformation, which is an entangling operator that presents a clear transformation relation. We also verified that the additivity property of the novel 3D EFrST is of a Fourier character by using its quantum mechanical description. As an application of this representation, the EFrST of the three-mode number state is calculated using the quantum description of the EFrST.
Based on the fact that the quantum mechanical version of Hankel transform kernel (the Bessel function) is just the transform between [Formula: see text] and [Formula: see text], two induced entangled state representations, and working with them, we derive fractional Fourier–Hankel transformation (FrFHT) caused by the operator [Formula: see text], where [Formula: see text] is named the core operator and is essential to the fractional transformation. The fractional property (additive rule) of the FrFHT can be explicitly proved.
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.