Development Of Quantum Phase Space Theory With The IWOP Technique And Entangled State Representations

The Quantum phase space theory becomes a Milestone from the Bohr-Sommerfeld quantization developing to the quasi-probability distribution function W q ,p of density operator proposed by Wigner. W q ,p avoids the problem of undefining of the q ,p function casued by Heisenberg uncertainty principle (we can not measure the position q and momentun p of the particle exactly), thus the Wigner function is not a real probability distribution. However, the marginal distributions of the Wigner function give the probability of measuring the positionq and momentun p of the particle respectively. By virtue of the technique of integration within anQ ordered product (IWOP) of operators, the paper study the representation, reconstruction of the quasi-probability distribution function and quantization of classical function, the relationship between classical transform and quantum unitary transform ect.. Our method is novel and unique. The main content of the paper is:1. By virtue of the IWOP techique, we reveal the essence of the Weyl ordering. By introducing the technique of integration within a Weyl ordered product (IWWOP) of operators, we obtain the Weyl ordered form of the Wigner operator which can be applied to develop the Quantum phase space theory. It can be seen from the article that the rule of the Weyl correspondence and the Wigner operator theory will be developed explicitly. On the other hand, we emphasize that for entangled particles one should treat their wave function as a whole, there is no physical meaning for an isolated particle's Wigner function, therefore thinking of entangled Wigner function (Wigner operator) is of necessity. Finally, we generalize the Weyl transform into entangled case.2. The coherent state can be represented by a small area in quantum phase space. We find that the move of the small area is controlled by the Liouville Theorem and is correspondence to the Fresnel transform in classical optics. Thus we can define the Fresne operator and discuss its association with the Wigner transform, i.e., there exists algorithmic isomorphism between ABCD transform of the Wigner distribution function and the optical Fresnel transform. We also find the kernel of the Fresnel transform is the inner product of the intermediate coordinate-momentum representation and coordinate representation, that is to say, we find the quantum correspondence of the Fresnel transform in classical optics. Additionally, we apply the Fresnel operator to discuss its relationship with the quantum tomograghy, and also to solve the Hamiltonian equation.Finally, we propose and construct the two-mode Fresnel operator and extend the above discussions to entangled case. The analogy of quantum mechanics and optics was pointed out by Schr?dinger. It is the analogy that he finds the idea of Schr?dinger equation. Our theory is consistent with Schr?dinger's ideas.3. We propose a new two-fold integration transform in q pphase space (we call it new transform in conveniece) which possesses some well-behaved transform properties and can be applied to finding the connection between the Weyl ordering and P Q (Q P ) ordering of operators. According to this new transform, we can obtain the fractional Fourier transform kernel from the chirplet function. We also develop this kind new transform to a more general case that can be further related to the transform between two mutually conjugate entangled state representations and study its properties and applications. We expect this transform could be implemented by experimentalists.4. By introducing the s-parameterized Wigner operator, we generalize the IWOP technique to the technique of integration within the s-ordered product of operators (IWSOP) which can unify the three integration techniques (within normal-, Weyl- and antinormal-ordering of operators) as one, and derive the s-ordered operator expansion formula of density operators. On the other hand, the essence of the s-parameterized quantization is discussed, the s-parameterized new transform is obtained and the photocount formula is also generalized into the s-parameterized case which can reduc to the usual one when s takes particular values. Our discussions provide greatly convenience for the study of the Quantum phase space theory. Finally, we extend the above discussions to the entangled case.5. By virtue of the IWOP technique, we correctly construct the tripartite entangled state representations (TESR) and investigate its quantum properties and some applications. We derive the three-mode squeezed operator and introduce the optical network for generating such an ideal tripartite entangled state. The three-mode entangled Wigner operator in this representation is constructed. Based on this form, we calculate the Wigner function expressions of some tripartite entangled states. The path integral formalism related to the tripartite entangled state is demonstrated. Additionally, the application of such entangled state in quantum teleportation and quantum Tomography thoery is analyzed as well. In the end, we introduce the three-mode optical entangled fractional Fourier transform (EFrFT) through two methods. The EFrFT, which is characteristic of the eigenmodes being three-variable Hermite polynomials, satisfies the additivity property. We also define two functions' convolution in the EFrFT scheme and obtain the convolution theorem using the TESR. The derivation is concise and rigorous because our calculation is based on Dirac's powerful representation theory.