Related papers: Remarks on quantum differential operators
This is a set of lecture notes for a graduate-level course on quantum algorithms, with an emphasis on quantum optimization algorithms. It is developed for applied mathematicians and engineers, and requires no previous background in quantum…
Spaces of differential forms over configuration spaces with Poisson measures are constructed. The corresponding Laplacians (of Bochner and de Rham type) on forms and associated semigroups are considered. Their probabilistic interpretation…
We investigate domain-wall/quantum field theory correspondences in various dimensions. We give particular emphasis to the special case of the quantum mechanics of 0--branes.
We have introduced q-analogues of bounded symmetric domains in our work q-alg/9703005. Given the simplest ones among those, the works q-alg/9603012 and math.QA/9803110 announce the relations describing the algebras of functions,…
Some ideas about phenomenological applications of quantum algebras to physics are reviewed. We examine in particular some applications of the algebras $U_ q (su_2)$ and $U_{qp}({\rm u}_2)$ to various dynamical systems and to atomic and…
The phenomenon of quantum entanglement is thoroughly investigated, focussing especially on geometrical aspects and on bipartite systems. After introducing the formalism and discussing general aspects, some of the most important separability…
Accurate models for open quantum systems -- quantum states that have non-trivial interactions with their environment -- may aid in the advancement of a diverse array of fields, including quantum computation, informatics, and the prediction…
A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis is given. The main result is that any operator with the above property must have a…
The nature of so-called differential-algebraic operators and their approximations is constitutive for the direct treatment of higher-index differential-algebraic equations. We treat first-order differential-algebraic operators in detail and…
We first review the properties of the conventional $\tau$-functions of the KP and Toda-lattice hierarchies. A straightforward generalization is then discussed. It corresponds to passing from differential to finite-difference equations; it…
An algebraic analysis framework for quantum calculus is proposed. The quantum derivative operator $D_{\tau ,\sigma}$ is based on two commuting bijections $\tau$ and $\sigma$ defined on an arbitrary set $M$ equipped with a tension structure…
I discuss some aspects of a lattice approach to canonical quantum gravity in a connection formulation, discuss how it differs from the continuum construction, and compare the spectra of geometric operators - encoding information about…
New concept of conditional differential invariant is discussed that would allow description of equations invariant with respect to an operator under a certain condition. Example of conditional invariants of the projective operator is…
A systematic exposition is given of the theory of invariant differential operators on a not necessarily reductive homogeneous space. This exposition is modelled on Helgason's treatment of the general reductive case and the special…
X states are a broad class of two-qubit density matrices that generalize many states of interest in the literature. In this work, we give a comprehensive account of various quantum properties of these states, such as entanglement,…
We study deterministic and quantum dynamics from a constructive "finite" point of view, since the introduction of a continuum, or other actual infinities in physics poses serious conceptual and technical difficulties, without any need for…
Cartan calculi on the extended quantum superplane are given. To this end, the noncommutative differential calculus on the extended quantum superplane is extended by introducing inner derivations and Lie derivatives.
We obtain necessary optimality conditions for higher-order infinite horizon problems of the calculus of variations via discrete quantum operators.
A very elementary introduction to quantum algebras is presented and a few examples of their physical applications are mentioned.
The relation between quantum systems associated to root systems and radial parts of Laplace operators on symmetric spaces is established. From this it follows the complete integrability of some quantum systems.