Related papers: Quantum groups: from Kulish-Reshetikhin discovery …
We develop a general theory of `quantum' diffeomorphism groups based on the universal comeasuring quantum group $M(A)$ associated to an algebra $A$ and its various quotients. Explicit formulae are introduced for this construction, as well…
A unified framework for different formulations of quantum theoery is introduced specifying what is meant by a quantum mechanical theory in general.
Quantum fields are shown to provide an example of infinite-dimensional quantum groups. A dictionary is established between quantum field and quantum group concepts: the expectation value over the vacuum is the counit, Wick's theorem is the…
We give a basic overview of computational complexity, query complexity, and communication complexity, with quantum information incorporated into each of these scenarios. The aim is to provide simple but clear definitions, and to highlight…
This paper surveys quantum learning theory: the theoretical aspects of machine learning using quantum computers. We describe the main results known for three models of learning: exact learning from membership queries, and Probably…
To any complex Hadamard matrix we associate a quantum permutation group. The correspondence is not one-to-one, but the quantum group encapsulates a number of subtle properties of the matrix. We investigate various aspects of the…
We classify the compact quantum groups $A_u(Q)$ (resp. $B_u(Q)$) up to isomorphism when $Q>0$ (resp. when $Q \bar{Q} \in {\mathbb R} I_n$). We show that the general $A_u(Q)$'s and $B_u(Q)$'s for arbitrary $Q$ have explicit decompositions…
A peculiar representation of the Lorentz group is suggested as a starting point for a consistent approach to relativistic quantum theory.
This article surveys quantum computational complexity, with a focus on three fundamental notions: polynomial-time quantum computations, the efficient verification of quantum proofs, and quantum interactive proof systems. Properties of…
Refined Algebraic Quantization and Group Averaging are powerful methods for quantizing constrained systems. They give constructive algorithms for generating observables and the physical inner product. This work outlines the current status…
In this paper, we have considered the problem of general conclusive quantum state classification; the necessary and sufficient conditions for the existence of conclusive classification strategies have also been presented. Moreover, we have…
Lecture notes from the 1994 CRM-CAP Summer School ``Particles and Fields '94''. Covers material written elsewhere in a more leisurely fashion, including many exercises. Describes derivation of quantum groups from the Chern-Simons lagrangian…
This thesis is a compilation of research in relativistic quantum information theory, and research in quantum reference frames. The research in the former category provides a fundamental construction of quantum information theory of…
Quantum planes and a new quantum cylinder are obtained as quantization of Poisson homogeneous spaces of two different Poisson structures on classical Euclidean group E(2).
The aim of this paper is to give a basic overview of Deformation Quantization (DQ) to physicists. A summary is given here of some of the key developments over the past thirty years in the context of physics, from quantum mechanics to…
Group convolutions and cross-correlations, which are equivariant to the actions of group elements, are commonly used in mathematics to analyze or take advantage of symmetries inherent in a given problem setting. Here, we provide efficient…
We present an algorithm for approximating linear categories of partitions (of sets). We report on concrete computer experiments based on this algorithm which we used to obtain first examples of so-called non-easy linear categories of…
We survey the existing parts of a classification of finite groups generated by orthogonal transformations in a finite-dimensional Euclidean space whose fixed point subspace has codimension one or two and extend it to a complete…
The notion of simple compact quantum group is introduced. As non-trivial (noncommutative and noncocommutative) examples, the following families of compact quantum groups are shown to be simple: (a) The universal quantum groups $B_u(Q)$ for…
In the present article we discuss the classification of quantum groups whose quasi-classical limit is a given simple complex Lie algebra $\mathfrak{g}$. This problem reduces to the classification of all Lie bialgebra structures on…