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The purpose of this paper is to introduce several basic theorems of coherent states and generalized coherent states based on Lie algebras su(2) and su(1,1), and to give some applications of them to quantum information theory for graduate students or non–experts who are interested in both Geometry and Quantum Information Theory. In the first half we make a… (More)
The renormalization group (RG) method as a powerful tool for reduction of evolution equations is formulated in terms of the notion of invariant manifolds. We start with derivation of an exact RG equation which is analogous to the Wilsonian RG equations in statistical physics and quantum field theory. It is clarified that the perturbative RG method… (More)
The aim of this paper is to give a hint for thinking to graduate or undergraduate students in Mathematical Physics who are interested in both Geometry and Quantum Computation. First I make a brief review of some properties on Grassmann manifolds and next I show a path between Grassmann manifolds and Quantum Computation which is related to the efficiency of… (More)
We make a brief review of (optical) Holonomic Quantum Computer (or Computation) proposed by Zanardi and Rasetti (quant–ph 9904011) and Pachos and Chountasis (quant–ph 9912093), and give a mathematical reinforcement to their works.
We give a possible generalization to the example in the paper of Zanardi and Rasetti (quant–ph 9904011). For this generalized one explicit forms of adiabatic connection, curvature and etc. are given.
Analysis of the WKB exactness in some homogeneous spaces is attempted. CP N as well as its noncompact counterpart D N,1 is studied. U(N + 1) or U(N, 1) based on the Schwinger bosons leads us to CP N or D N,1 path integral expression for the quantity, tre −iHT , with the aid of coherent states. The WKB approximation terminates in the leading order and yields… (More)
U(N) coherent states over Grassmann manifold, G N,n ≃ U(N)/(U(n) × U(N − n)), are formulated to be able to argue the WKB-exactness, so called the localization of Duistermaat-Heckman, in the path integral representation of a character formula. The exponent in the path integral formula is proportional to an integer k that labels the U(N) representation. Thus… (More)
In this paper we consider a model of quantum computation based on n atoms of laser–cooled and trapped linearly in a cavity and realize it as the n atoms Tavis– Cummings Hamiltonian interacting with n external (laser) fields. We solve the Schrödinger equation of the model in the case of n=2 and construct the controlled NOT gate by making use of a resonance… (More)
In this paper we treat a cavity QED quantum computation. Namely, we consider a model of quantum computation based on n atoms of laser–cooled and trapped linearly in a cavity and realize it as the n atoms Tavis–Cummings Hamiltonian interacting with n external (laser) fields. We solve the Schrödinger equation of the model in the weak coupling regime to… (More)