Quantum theory: Concepts and methods

Introduction.- The Source of Examples.- Invariance Equation in the Differential Form.- Film Extension of the Dynamics: Slowness as Stability.- Entropy, Quasi-Equilibrium and Projector Field.- Newton Method with Incomplete Linearization.- Quasi-chemical Representation.- Hydrodynamics from Grad's Equations: Exact Solutions.- Relaxation Methods.- Method of Invariant Grids.- Method of Natural Projector.- Geometry of Irreversibility: The Film of Nonequilibrium States.- Slow Invariant Manifolds for Open Systems.- Estimation of Dimension of Attractors.- Accuracy Estimation and Post-Processing.- Conclusion.

http://www.math.le.ac.uk/people/ag153/homepage/Beginning.pdf

Invariant Manifolds for Physical and Chemical Kinetics

This is the first comprehensive introduction to the powerful moment approach for solving global optimization problems (and some related problems) described by polynomials (and even semi-algebraic functions). In particular, the author explains how to use relatively recent results from real algebraic geometry to provide a systematic numerical scheme for computing the optimal value and global minimizers. Indeed, among other things, powerful positivity certificates from real algebraic geometry allow one to define an appropriate hierarchy of semidefinite (SOS) relaxations or LP relaxations whose optimal values converge to the global minimum. Several extensions to related optimization problems are also described. Graduate students, engineers and researchers entering the field can use this book to understand, experiment with and master this new approach through the simple worked examples provided.

An Introduction to Polynomial and Semi-Algebraic Optimization

This is a collection of references (papers, books, preprints, book reviews, Ph. D. thesis, patents, web sites, etc.), sorted alphabetically and (some of them) classified by subject, on foundations of quantum mechanics and quantum information. Specifically, it covers hidden variables (``no-go'' theorems, experiments), interpretations of quantum mechanics, entanglement, quantum effects (quantum Zeno effect, quantum erasure, ``interaction-free'' measurements, quantum ``non-demolition'' measurements), quantum information (cryptography, cloning, dense coding, teleportation), and quantum computation.

/pdf/bibliographic-guide-to-the-foundations-of-quantum-mechanics-5bg1n4o4cz.pdf

Bibliographic guide to the foundations of quantum mechanics and quantum information

We compute the volume of the (N2 − 1)-dimensional set N of density matrices of size N with respect to the Bures measure and show that it is equal to that of an (N2 − 1)-dimensional hyper-hemisphere of radius 1/2. For N = 2 we obtain the volume of the Uhlmann hemisphere, ½S3 ⊂ 4. We find also the area of the boundary of the set N and obtain analogous results for the smaller set of all real density matrices. An explicit formula for the Bures–Hall normalization constants is derived for an arbitrary N.

https://chaos.if.uj.edu.pl/~karol/pdf/SZ03.pdf

Bures volume of the set of mixed quantum states

Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

Geometrical aspects of quantum computing are reviewed elementarily for nonexperts and/or graduate students who are interested in both geometry and quantum computation. 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. An interesting current topic of holonomic quantum computation is also covered. Also, some related advanced topics are discussed.

/pdf/introduction-to-grassmann-manifolds-and-quantum-computation-3ekdo1gu05.pdf

Introduction to Grassmann manifolds and quantum computation

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}

/pdf/introduction-to-coherent-states-and-quantum-information-3apygbmyn6.pdf

Introduction to Coherent States and Quantum Information Theory

Using the generalized coherent states it is shown that the path integral formulas for SU(2) and SU(1,1) (in the discrete series) are WKB exact, if it is started from the trace of e−iTĤ, where H is given by a linear combination of generators. In this case, the WKB approximation is achieved by taking a large ‘‘spin’’ limit: J,K→∞, under which it is found that each coefficient vanishes except the leading term which indeed gives the exact result. It is further pointed out that the discretized form of path integral is indispensable, in other words, the continuum path integral expression sometimes leads to a wrong result. Therefore great care must be taken when some geometrical action would be adopted, even if it is so beautiful as the starting ingredient of path integral. Discussions on generalized coherent states are also presented both from geometrical and simple oscillator (Schwinger boson) points of view.

Coherent states, Path integral, and Semiclassical approximation

U(N) coherent states over Grassmann manifold, GN,n≂U(N)/(U(n)×U(N−n)), are formulated to be able to argue the WKB exactness in the path integral representation of a character formula. The phenomena is the so‐called localization of Duistermaat–Heckman. The exponent in the path integral formula is proportional to an integer k labeling the U(N) representation. Thus, when k→∞ a usual semiclassical approximation, by regarding k∼1/ℏ, can be performed to yield a desired conclusion. The mechanism of the localization is uncovered by the help of the (generalized) Schwinger boson technique. The discussion on the Feynman kernel is also presented.

Kazuyuki Fujii

Papers

Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

Introduction to Grassmann manifolds and quantum computation

Introduction to Coherent States and Quantum Information Theory

Coherent states, Path integral, and Semiclassical approximation

Coherent states over Grassmann manifolds and the WKB exactness in path integral