arXiv subjects
Publications and source records attributed to Kazuyuki Fujii.
We give a modern approach to the famous Cardano and Ferrari formulas in the algebraic equations with three and four degrees. Namely, we reconstruct these formulas from the point of view of superposition principle in quantum computation based on three and four level systems which are being developed by the author. We also present a problem on some relation between Galois theory and Qudit theory.
A set of generators of generalized Pauli matrices play a crucial role in quantum computation based on n level systems of an atom. In this paper we show how to construct them by making use of Rabi oscillations. We also construct the generalized Walsh-Hadamard matrix in the case of three level systems and present some related problems.
In this paper we study the behavior of (laser--cooled) m--atoms trapped in a cavity interacting with a photon $...$ Cavity QED $...$ and attempt to solve the Schr{ö}dinger equation of this model in the strong coupling regime. In the case of m = 2 we construct Bell--Schr{\" o}dinger cat states (in our terminology) and obtain with such bases some unitary transformations by making use of the rotating wave approximation under new resonance conditionscontaining the Bessel functions, which will be applied to construct important quantum logic gates in Quantum Computation. Moreover we propose in the case of m = 3 a crucial problem to solve on Quantum Computation.
In this paper we consider a model of an atom with n energy levels interacting with n(n-1)/2 external (laser) fields which is a natural extension of two level system, and assume the rotating wave approximation (RWA) from the beginning. We revisit some construction of analytical solutions (which correspond to Rabi oscillations) of the model in the general case and examine it in detail in the case of three level system.
In this paper we propose a Hamiltonian generalizing the interaction of the two--level atom and both the single radiation mode and external field $...$ a kind of cavity QED. We solve the Schrodinger equation in the strong coupling regime by making use of rotating wave approximation under new resonance conditions containing the Bessel functions and etc, and obtain unitary transformations of four types corresponding to Rabi oscillations which perform quantum logic gates in Quantum Computation.
In this paper we propose some Hamiltonian characterizing the interaction of the two-level atom and both the single radiation mode and external field, which might be a generalization of that of Sch{ö}n and Cirac (quant-ph/0212068). We solve them in the strong coupling regime under some conditions (the rotating wave approximation, resonance condition and etc), and obtain unitary transformations of four types to perform Quantum Computation.
In this paper we propose a Hamiltonian of the n-level system by making use of generalized Pauli matrices.
In this paper we treat the 2--level system interacting with external fields without the rotating wave approximation and construct some approximate solutions in the strong coupling regime.
In this paper we consider some (generalized) models which deal with n--level atom interacting with a single radiation mode and study a general structure of Rabi floppings of these generalized ones in the strong coupling regime. To solve these models we introduce multi cat states of Schr{\" o}dinger in our terminology and extend the result in (quant--ph/0203135) which Rabi frequencies are given by matrix elements of generalized coherent operators (quant--ph/0202081) under the rotating--wave approximation. Our work is a full generalization of Frasca (quant--ph/0111134) and Fujii (quant--ph/0203135). In last we make a brief comment on an application to Quantum Computation on the space of qudits.
In this paper we generalize the Jaynes--Cummings Hamiltonian by making use of some operators based on Lie algebras su(1,1) and su(2), and study a mathematical structure of Rabi floppings of these models in the strong coupling regime. We show that Rabi frequencies are given by matrix elements of generalized coherent operators (quant--ph/0202081) under the rotating--wave approximation. In the first half we make a general review of coherent operators and generalized coherent ones based on Lie algebras su(1,1) and su(2). In the latter half we carry out a detailed examination of Frasca (quant--ph/0111134) and generalize his method, and moreover present some related problems. We also apply our results to the construction of controlled unitary gates in Quantum Computation. Lastly we make a brief comment on application to Holonomic Quantum Computation.
We construct the exchange gate with small elementary gates on the space of qudits, which consist of three controlled shift gates and three "reverse" gates. This is a natural extension of the qubit case. We also consider a similar subject on the Fock space, but in this case we meet with some different situation. However we can construct the exchange gate by making use of generalized coherent operator based on the Lie algebra su(2) which is a well--known method in Quantum Optics. We moreover make a brief comment on "imperfect clone".
In the first half we make a short review of coherent states and generalized coherent ones based on Lie algebras su(2) and su(1,1), and the Schwinger's boson method to construct representations of the Lie algebras. In the second half we make a review of recent developments on both swap of coherent states and cloning of coherent states which are important subjects in Quantum Information Theory.
Geometrical aspects of quantum computing are reviewed elementarily for non-experts and/or graduate students who are interested in both Geometry and Quantum Computation. In the first half we show how to treat Grassmann manifolds which are very important examples of manifolds in Mathematics and Physics. Some of their applications to Quantum Computation and its efficiency problems are shown in the second half. An interesting current topic of Holonomic Quantum Computation is also covered. In the Appendix some related advanced topics are discussed.
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 general review of coherent states and generalized coherent states based on Lie algebras su(2) and su(1,1) from the geometric point of view and, in particular, prove that each resolution of unity can be obtained by the curvature form of some bundle on the parameter space. In the latter half we apply a method of generalized coherent states to some important topics in Quantum Information Theory, in particular, swap of coherent states and cloning of coherent ones. We construct the swap operator of coherent states by making use of a generalized coherent operator based on su(2) and show an "imperfect cloning" of coherent states, and moreover present some related problems. In conclusion we state our dream, namely, a construction of {\bf Geometric Quantum Information Theory}.
This is a continuation of the preceding paper (hep-ph/0108219). First of all we make a brief review of generalized coherent states based on Lie algebra su(1,1) and prove that the resolution of unity can be obtained by the curvature form of some bundle. Next for a set of generalized coherent states we define a universal bundle over the infinite-dimensional Grassmann manifold and construct the pull-back bundle making use of a projector from the parameter space to this Grassmann one. We mainly study Chern characters of these bundles. Although the Chern characters in the infinite-dimensional case are in general not easy to calculate, we can perform them for the special cases. In this paper we report our calculations and propose some problems.
First we make a brief review of coherent states and prove that the resolution of unity can be obtained by the 1-st Chern character of some bundle. Next we define a Grassmann manifold for a set of coherent states and construct the pull-back bundle making use of a projector from the parameter space to this Grassmann manifold. We study some geometric properties (Chern-characters mainly) of these bundles. Although the calculations of Chern-characters are in general not easy, we can perform them for the special cases. In this paper we report our calculations and propose some interesting problems to be solved in the near future.
The aim of this paper is to introduce our idea of Holonomic Quantum Computation (Computer). Our model is based on both harmonic oscillators and non-linear quantum optics, not on spins of usual quantum computation and our method is moreover completely geometrical. We hope that therefore our model may be strong for decoherence.
We make a brief comment on measurement of quantum operators with degenerate eigenstates and apply to quantum teleportation. We also try extending the quantum teleportation by Bennett et al [5] to more general situation by making use of generalized Bell states.