Symbolic Package for Quantum Groups, Noncommutative Algebras, and Logics

This paper describes packages for symbolic calculations in quantum groups, noncommutative differential geometry, and multivalued logic. The package for quantum groups and the program for logic are written in Mathematica 3.0 and/or 4.0. As an example, some results in the logic obtained using these packages are presented.     

On fairness,full cooperation,and quantum game with incomplete information

Quantum entanglement has emerged as a new resource to enhance cooperation and remove dilemmas. This paper aims to explore conditions under which full cooperation is achievable even when the information of payoff is incomplete.Based on the quantum version of the extended classical cash in a hat game, we demonstrate that quantum entanglement may be used for achieving full cooperation or avoiding moral hazards with the reasonable profit distribution policies even when the profit is uncertain to a certain degree. This research further suggests that the fairness of profit distribution should play an important role in promoting full cooperation. It is hopeful that quantum entanglement and fairness will promote full cooperation among distant people from various interest groups when quantum networks and quantum entanglement are accessible to the public.     

Reaction-Diffusion Processes, Critical Dynamics, and Quantum Chains

The master equation describing non-equilibrium one-dimensional problems like diffusion limited reactions or critical dynamics of classical spin systems can be written as a Schrödinger equation in which the wave function is the probability distribution and the Hamiltonian is that of a quantum chain with nearest neighbor interactions. Since many one-dimensional quantum chains are integrable, this opens a new field of applications. At the same time physical intuition and probabilistic methods bring new insight into the understanding of the properties of quantum chains. A simple example is the asymmetric diffusion of several species of particles which leads naturally to Hecke algebras and q-deformed quantum groups. Many other examples are given. Several relevant technical aspects like critical exponents, correlation functions, and finite-size scaling are also discussed in detail.     

Universal R Matrix of Two-Parameter Deformed Quantum Group Uqs(SU(1, 1))

The universal R matrix of the two-parameterdeformed quantum group U_qs(SU(1, 1)) isderived. In previous work we suggested a method toderive the universal R matrix of the two-parameterdeformed quantum group U_qs(SU(2)). This method isdifferent from that of the quantum double; it is simpleand efficient for quantum groups of low rank at least.This paper studies the universal R matrix of thetwo-parameter deformed quantum group U_qs(SU(1, 1))using the same approach.     

Coxeter groups A4, B4 and D4 for two-qubit systems

The Coxeter–Weyl groups W(A₄), W(B₄), and W(D₄) have proven very useful for two-qubit systems in quantum information theory. A simple technique is employed to construct the unitary matrix representations of the groups, based on quaternionic transformation of the usual reflection matrices. The von Neumann entropy of each reduced density matrix is calculated. It is shown that these unitary matrix representations are naturally related to various universal quantum gates and they lead to entangled states. Canonical decomposition of generators in terms of fundamental gate representations is given to construct the quantum circuits.     

Generalized Ehrenfest Relations,Deformation Quantization,and the Geometry of Inter-model Reduction

This study attempts to spell out more explicitly than has been done previously the connection between two types of formal correspondence that arise in the study of quantum–classical relations: one the one hand, deformation quantization and the associated continuity between quantum and classical algebras of observables in the limit \(\hbar \rightarrow 0\), and, on the other, a certain generalization of Ehrenfest’s Theorem and the result that expectation values of position and momentum evolve approximately classically for narrow wave packet states. While deformation quantization establishes a direct continuity between the abstract algebras of quantum and classical observables, the latter result makes in-eliminable reference to the quantum and classical state spaces on which these structures act—specifically, via restriction to narrow wave packet states. Here, we describe a certain geometrical re-formulation and extension of the result that expectation values evolve approximately classically for narrow wave packet states, which relies essentially on the postulates of deformation quantization, but describes a relationship between the actions of quantum and classical algebras and groups over their respective state spaces that is non-trivially distinct from deformation quantization. The goals of the discussion are partly pedagogical in that it aims to provide a clear, explicit synthesis of known results; however, the particular synthesis offered aspires to some novelty in its emphasis on a certain general type of mathematical and physical relationship between the state spaces of different models that represent the same physical system, and in the explicitness with which it details the above-mentioned connection between quantum and classical models.     

Yangians,integrable quantum systems and Dorey's rule

It was pointed out by P. Dorey that the three-point couplings between the quantum particles in affine Toda field theories have a remarkable Lie-theoretic interpretation. It is also well known that such theories admit quantum affine algebras as quantum symmetry groups, and widely believed that the quantum particles correspond to the so-called fundamental representations of these algebras. This led to the conjecture that Dorey's rule should describe when a fundamental representation occurs with non-zero multiplicity in a tensor product of two other fundamental representations. The purpose of this paper is to prove this conjecture, both for quantum affine algebras and for Yangians. The result reveals a hitherto unsuspected role played by Coxeter elements (and their twisted analogues) in the representation theory of these algebras.     

CCAP for Universal Discrete Quantum Groups

We show that the discrete duals of the free orthogonal quantum groups have the Haagerup property and the completely contractive approximation property. Analogous results hold for the free unitary quantum groups and the quantum automorphism groups of finite-dimensional C^*-algebras. The proof relies on the monoidal equivalence between free orthogonal quantum groups and SU _q (2) quantum groups, on the construction of a sufficient supply of bounded central functionals for SU _q (2) quantum groups, and on the free product techniques of Ricard and Xu. Our results generalize previous work in the Kac setting due to Brannan on the Haagerup property, and due to the second author on the CCAP.     

The Legendre Structure of the Parisi Formula

We study easy quantum groups, a combinatorial class of orthogonal quantum groups introduced by Banica–Speicher in 2009. We show that there is a countable descending chain of easy quantum groups interpolating between Bichon’s free wreath product with the permutation group S_n and a semi-direct product of a permutation action of S_n on a free product. This reveals a series of new commutation relations interpolating between a free product construction and the tensor product. Furthermore, we prove a dichotomy result saying that every hyperoctahedral easy quantum group is either part of our new interpolating series of quantum groups or belongs to a class of semi-direct product quantum groups recently studied by the authors. This completes the classification of easy quantum groups. We also study combinatorial and operator algebraic aspects of the new interpolating series.     

R-Matrices for Some Inhomogeneous Quantum Groups

In this paper, some new generalized inhomogeneous quantum groups corresponding to the homogeneous multiparameter quantum groups GL_X,qij(N)are constructed, furthermore,the R-matrices for these inhomogeneous quantum groups are found.     

Non-Semi-Regular Quantum Groups Coming from Number Theory

 In this paper, we study C^*-algebraic quantum groups obtained through the bicrossed product construction. Examples using groups of adeles are given and they provide the first examples of locally compact quantum groups which are not semi-regular: the crossed product of the quantum group acting on itself by translations does not contain any compact operator. We describe all corepresentations of these quantum groups and the associated universal C^*-algebras. On the way, we provide several remarks on C^*-algebraic properties of quantum groups and their actions. Received: 10 October 2002 / Accepted: 10 October 2002 Published online: 24 January 2003 Communicated by A. Connes     

Incorporating fluorescent quantum dots into water-soluble polymer

The fluorescent quantum dot-polymer composites were fabricated by incorporating thioglycolic acid capped CdTe quantum dots into polyacrylamide via cross-linking agents. The CdTe-polyacrylamide composites were characterized by fluorescence spectrophotometer and fluorescence microscope. The result shows that the quantum dot-polymer composites show strong photoluminescence in aqueous solution. The photoluminescence spectrum of quantum dot-polymer composites exhibits a slight blue shift compared to that of initial CdTe quantum dots. The slight shift might be attributed to the covalently bonding between the carboxyl groups of thiolglycolic acid capped on CdTe quantum dots and the amide groups of the polyacrylamide chains.     

A short introduction to quantum symmetry

In quantum theory, symmetries more general than groups are possible. We give a general definition of a quantum symmetry, such that symmetry operations act on the Hilbert space of physical states and notions of unitarity, invariance and covariance are defined. Within this frame, weak quasi quantum groups are described as a natural generalization of group algebras. Consistency with locality distinguishes them from more general quantum symmetries. To find the new kinds of symmetry one should investigate low dimensional quantum systems such as two-dimensional layers.     

Quantum Public-Key Cryptosystem

Quantum one-way functions play a fundamental role in cryptography because of its necessity for the secure encryption schemes taking into account the quantum computer. In this paper our purpose is to establish a theoretical framework for a candidate of the quantum one-way functions and quantum trapdoor functions based on one-parameter unitary groups. The dynamics of parameterized unitary groups ensure the one-wayness and quantum undistinguishability in different levels, and the physical feasibility are derived from the simultaneous approximation of its infinitesimal generators. Moreover, these special functions are used to construct new cryptosystems-the quantum public-key cryptosystems for encrypting both the classical and quantum information.     

Relations betweenGL p,q (2)'s

Quantum groups have some peculiar properties is two dimensions. We formulate conditions sufficient for the product of two quantum matrices (with not necessarily the same values of deformation parameters) to be a quantum matrix again. This is then used to study the powers and exponential form of matrices fromGL _p,q (2), generalising this way properties ofGL _q (2)-matrices.     

Gauge theories, vertex models, and quantum groups

It is known that the Jones polynomial of knot theory, and its generalizations, are closely related to the integrable “vertex models” of two-dimensional statistical mechanics, and to quantum groups. In this paper, an attempt is made to show on a priori grounds, starting only from general covariance of three-dimensional Chern-Simons gauge theory and two-dimensional “duality”, why this must be so.     

Random sets,q-distributions and quantum groups

We have shown that the non-extensivity of classical set theory is related to unitary quantum groups. Using this non-extensivity property, we define a q-distribution, a binomial q-distribution and a Poisson q-distribution.     

A remark on quasi-triangular Lie algebras

Quasi-triangular quantum Lie algebras are algebras of quantum Lie derivatives. We show how to associate a quasi-triangular quantum Lie algebra to any quantum group defined by exchange relations. This provides a systematical way for constructing bi-covariant differential calculus on exchange quantum groups.