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Chinese Physics B

New theorem relating two-mode entangled tomography to two-mode Fresnel operator

Chuanmei Xie, Hong-Yi Fan

Published Jan 1, 2012• 2 citations
Publisher

Abstract

Based on the Fan—Hu's formalism, i.e., the tomogram of two-mode quantum states can be considered as the module square of the states' wave function in the intermediate representation, which is just the eigenvector of the Fresnel quadrature phase, we derive a new theorem for calculating the quantum tomogram of two-mode density operators, i.e., the tomogram of a two-mode density operator is equal to the marginal integration of the classical Weyl correspondence function of F 2 † ρ F 2 , where F 2 is the two-mode Fresnel operator. An application of the theorem in evaluating the tomogram of an optical chaotic field is also presented.

DOI
10.1088/1674-1056/21/1/010302

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Based on the Fan—Hu's formalism, i.e., the tomogram of two-mode quantum states can be considered as the module square of the states' wave function in the intermediate representation, which is just the eigenvector of the Fresnel quadrature phase, we derive a new theorem for calculating the quantum tomogram of two-mode density operators, i.e., the tomogram of a two-mode density operator is equal to the marginal integration of the classical Weyl correspondence function of F 2 † ρ F 2 , where F 2 is the two-mode Fresnel operator. An application of the theorem in evaluating the tomogram of an optical chaotic field is also presented.
According to Fan—Hu's formalism (Fan Hong-Yi and Hu Li-Yun 2009 Opt. Commun . 282 3734 ) that the tomogram of quantum states can be considered as the module-square of the state wave function in the intermediate coordinate—momentum representation which is just the eigenvector of the Fresnel quadrature phase, we derive a new theorem for calculating quantum tomogram of density operator, i.e., the tomogram of a density operator ρ is equal to the marginal integration of the classical Weyl correspondence function of F † ρ F , where F is the Fresnel operator. Applications of this theorem to evaluating the tomogram of optical chaotic field and squeezed chaotic optical field are presented.
Similar in spirit to the preceding work 9 Fan, H-Y and Hu, L-Y. 2009. Opt. Commun., 282: 3734–3736. [Crossref], [Web of Science ®] , [Google Scholar], where the relation between optical Fresnel transformation and quantum tomography is revealed, we explore this kind of relationship in a two-mode entangled case. We show that under the two-mode Fresnel transformation the entangled state density becomes density operator , which is just the Radon transform of the two-mode Wigner operator in entangled form, i.e. where F 2 is a two-mode Fresnel operator in quantum optics, and s , r are the complex-value expressions of (A,B,C,D). So the probability distribution is the tomography (Radon transform of the two-mode Wigner function), and gives a tomogram. Similarly, we find where is the conjugated state to .
We show that for tomographic approach there exist two mutual conjugate quantum states | p ,σ,τ⟩ and | x ,μ,ν⟩ (named the intermediate coordinate-momentum representation), and the two Radon transforms of the Wigner operator are just the pure-state density matrices | p ⟩ σ,τσ,τ ⟨ p | and | x ⟩ λ,νλ,ν ⟨ x | respectively. As a result, the tomogram of quantum states can be considered as the module-square of the states' wave function in these two representations. Throughout the paper we fully employ the technique of integration within an ordered product of operators. In this way we establish a new convenient formalism of quantum tomogram.
By searching for Q- and P-ordering form of the coordinate-momentum intermediate representation |x⟩λ,νλ,ν⟨x| (formed by the eigenvector of λQ+νP with eigenvalue being x, Q,P=iħ), and noting that |x⟩λ,νλ,ν⟨x|=δ(x-λQ-νP) is just the Radon transformation of the Wigner operator, we find new relationship between a quantum state ψ’s tomogram and its wave function in coordinate (or momentum) representation, i.e. the tomogram can be split into two parts, one is ψq’s Gaussian integration transformation with the parameter ν and another is ψ∗p’s Gaussian integration transformation with the parameter λ. In this way, the quantum tomogram of number state is conveniently deduced. We also derive Radon transform of δ(p-P)δ(q-Q) and δ(q-Q)δ(p-P), which can be either viewed as P- and Q-ordered operator correspondence of the classical function δ(x-λq-νp).