A class of quantum channels and completely positive maps (CPMs) are introduced and investigated. These, which we call subspace preserving (SP) CPMs has, in the case of trace preserving CPMs, a simple interpretation as those which preserve probability weights on a given orthogonal sum decomposition of the Hilbert space of a quantum system. Several equivalent characterizations of SP CPMs are prov...

A major challenge in the training of recurrent neural networks is the so-called vanishing or exploding gradient problem. The use of a norm-preserving transition operator can address this issue, but parametrization is challenging. In this work we focus on unitary operators and describe a parametrization using the Lie algebra u(n) associated with the Lie group U(n) of n× n unitary matrices. The e...

The group of area preserving diffeomorphisms of the annulus acts on its Lie algebra, the globally Hamiltonian vectorfields on the annulus. We consider a certain Hilbert space completion of this group (thinking of it as a group of unitary operators induced by the diffeomorphisms), and prove that the projection of an adjoint orbit onto a "Cartan" subalgebra isomorphic to Lz([0, 1]) is an infinite...

The unitary Cayley graph Xn has vertex set Zn = {0, 1,…, n-1} and vertices u and v are adjacent, if gcd(uv, n) = 1. In [A. Ilić, The energy of unitary Cayley graphs, Linear Algebra Appl. 431 (2009) 1881–1889], the energy of unitary Cayley graphs is computed. In this paper the Wiener and hyperWiener index of Xn is computed.

We consider the graph whose vertex set is a conjugacy class C consisting of finite-rank self-adjoint operators on complex Hilbert space H. The dimension H assumed to be not less than 3. In case when from have two eigenvalues only, we obtain Grassmann formed by k-dimensional subspaces H, where k smallest eigenspaces. Chow's theorem describes automorphisms this for k>1. Under assumption that more...

Abstract In this paper we consider real or complex skew-Hamiltonian/Hamiltonian pencils λS −H, i.e., pencils where S is a skew-Hamiltonian and H is a Hamiltonian matrix. These pencils occur for example in the theory of continuous time, linear quadratic optimal control problems. We reduce these pencils to canonical and Schur-type forms under structure-preserving transformations, i.e., J-congruen...

Given commuting families of Hermitian matrices {A1, . . . , Ak} and {B1, . . . , Bk}, conditions for the existence of a completely positive map Φ, such that Φ(Aj) = Bj for j = 1, . . . , k, are studied. Additional properties such as unital or / and trace preserving on the map Φ are also considered. Connections of the study to dilation theory, matrix inequalities, unitary orbits, and quantum inf...

We present a scheme for achieving coherent spin squeezing of nuclear spin states in semiconductor quantum dots. The nuclear polarization dependence of the electron spin resonance generates a unitary evolution that drives nuclear spins into a collective entangled state. The polarization dependence of the resonance generates an area-preserving, twisting dynamics that squeezes and stretches the nu...

Abstract We propose a novel matrix regularization for tensor fields. In this regularization, fields are described as rectangular matrices, and area-preserving diffeomorphisms local rotations of the orthonormal frame both realized unitary similarity transformations matrices in unified way. also show that commutator corresponds to covariantized Poisson bracket large-N limit.