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Journal of Mathematical Physics

A novel 3D entangled wavelet transformation and its properties

Shoukang Chang, Liyun Hu

Published Oct 1, 2019• 1 citation
Publisher

Abstract

Based on the quantum mechanics representation, we propose a novel 3D entangled wavelet transformation. This kind of three entangled state representation can be generated theoretically by using a two-cascaded beam splitter, which has the completeness and orthogonal property. Some properties of 3D WTs are discussed, including the Parseval theory of 3D WTs, its inverse formula, and the orthogonality of the mother wavelets in parameter space. In addition, the mother wavelet condition for the 3D WTs is also derived.

DOI
10.1063/1.5102060

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Based on the quantum mechanics representation, we propose a novel 3D entangled wavelet transformation. This kind of three entangled state representation can be generated theoretically by using a two-cascaded beam splitter, which has the completeness and orthogonal property. Some properties of 3D WTs are discussed, including the Parseval theory of 3D WTs, its inverse formula, and the orthogonality of the mother wavelets in parameter space. In addition, the mother wavelet condition for the 3D WTs is also derived.
Beam splitter,as a kind of linear optics instruments,has many applications such as in quantum optics and quantum information,including the preparation of nonclassical quantum states and entangled state representation.In Heisenberg picture,on the one hand,the relation of input-output of beam splitter can be easily obtained.Especially for the multicascaded beam-splitters,the input-output relation can also be directly obtained by the input-output relation of single beam splitter.On the other hand,we often need to calculate the probabilities of detecting photon number in many cases,thus we need to turn into Schrdinger picture for simplifying our calculation.Based on the equivalence between both pictures,the relation between transformation matrixes connecting these two pictures is derived.That is to say, the transform matrix corresponding to the Schrdinger picture can be obtained by transposing the transform matrix in Heisenberg picture.This concise relation constructs a bridge connecting two pictures and simplifies our calculation in the Schrdinger picture rather than step by step.Using the relation between transform matrixes of both pictures and combining the technique of integration within ordered product of operator,we further consider the coordination representation,normally ordering form and exponential expression of single beam-splitter.Then we further examine the coordination representation,normally ordering form and exponential expression of two-cascaded beam-splitters.As a generalization,the method is extended to the case of multi-cascaded beam-splitters.These investigations provide an effective way to prepare multi-mode entangled states and qubit states.In addition,a general method is shown of obtaining the total operator and its normally ordering form as well as Schmidt decomposition of the linear systems consisting of beam-splitters.As applications,2-cascaded beam-splitters is used to generate a new quantum mechanics representation and prepare the qubit states with the help of conditional measurement.The Schmidt decomposition of three-mode entangled state representation can be directly obtained by the coordination representation of 2-cascaded beam-splitters,which shows the property of entanglement.In addition,based on this representation we can clearly see that when the input states of first beam splitter are two coordinate states,the output states cannot be entangled.This implies that although the coordinate states are nonclassical,the entangled state can not be prepared either.The new proposed quantum mechanics representation will be further used to investigate the optical transformations,including wavelet transformation,Fourier transform,fractional Fourier transform,et al.Therelevant discussion will be our aim in the future research.
A new kind of tripartite coherent-entangled state (CES) ∣β,γ,x⟩μντ\ket{β,γ, x}_{μντ}∣β,γ,x⟩μντ​ is proposed, which exhibits the properties of both coherence and entanglement. We investigate its completeness and orthogonality, and find it can make up a representation of tripartite CES. A protocol for generating the tripartite CES is proposed using asymmetric beam splitter. Applications of the tripartite CES in quantum optics are also presented.