We introduce positive operator-valued measure (POVM) generated by the projective unitary representation of a direct product locally compact Abelian group G with its dual $${\hat{G}}$$ . The method is based upon Pontryagin duality allowing to establish an isometrical isomorphism between space Hilbert–Schmidt operators in $$L^2(G)$$ and Hilbert $$L^2({\hat{G}}\times G)$$ Any such determines pair ...

“Quantum holonomy” as defined by Berry and Simon, and based on the parallel transport of Bott and Chern, can be considerably extended. There is a natural “parallelity” W*dW = (dW)* W within the Hilbert--Schmidt operators W This defines parallelity and holonomy along curves of density operators e = WW*. There is an intrinsic nonlinearity in the parallel transport which dissolves for curves of pr...

Information-theoretic aspects of quantum inseparability of mixed states are investigated in terms of the α-entropy inequalities and teleportation fidelity. Inseparability of mixed states is defined and a complete characterization of the inseparable 2 × 2 systems with maximally disordered subsystems is presented within the Hilbert-Schmidt space formalism. A connection between teleportation and n...

We derive the correspondence between regularization operators used in Regularization Networks and Hilbert Schmidt Kernels appearing in Support VectorMachines. More specifically, we prove that the Green’s Functions associated with regularization operators are suitable Support Vector Kernels with equivalent regularization properties. As a by–product we show that a large number of Radial Basis Fun...

We obtain two sided estimates for the Bures volume of an arbitrary subset of the set of N ×N density matrices, in terms of the Hilbert-Schmidt volume of that subset. For general subsets, our results are essentially optimal (for large N). As applications, we derive in particular nontrivial lower and upper bounds for the Bures volume of sets of separable states and for sets of states with positiv...

In this note we study some basic properties of the γinterpolation method introduced in [6] e.g. interpolation of Bochner spaces and interpolation of analytic operator valued functions in the sense of Stein. As applications we consider the interpolation of almost summing operators (in particular γ-radonifying and Hilbert-Schmidt operators) and of γ-Sobolev spaces as introduced in [6]. We also co...

This paper is concerned with a technique for solving Volterra integral equations in the reproducing kernel Hilbert space. In contrast with the conventional reproducing kernel method,the Gram-Schmidt process is omitted here and satisfactory results are obtained. The analytical solution is represented in the form of series. An iterative method is given to obtain the approximate solution. The conv...

Abstract. In this paper we obtain a necessary and sufficient condition for the canonical solution operator to ∂ restricted to radial symmetric Bergman spaces to be a Hilbert-Schmidt operator. We also discuss compactness of the solution operator in spaces of entire functions in one variable. In the sequel we consider several examples and also treat the case of weighted spaces of entire functions...

We propose an operational measure of distance of two quantum states, which conversely tells us their closeness. This is defined as a sum of differences in partial knowledge over a complete set of mutually complementary measurements for the two states. It is shown that the measure is operationally invariant and it is equivalent to the Hilbert-Schmidt distance. The operational measure of distance...

In these lectures we shall present an introduction of the theory of stochastic integration in UMD Banach spaces and some of its applications. The Hilbert space approach to stochastic partial differential equations (SPDEs) was pioneered in the 1980s by Da Prato and Zabczyk. Under suitable Lipschitz conditions, mild solutions of semilinear SPDEs in Hilbert spaces can be obtained by solving a fixe...