Related papers: Classical and nonclassical randomness in quantum m…
Recently, weak measurements have attracted a lot of interest as an experimental method for the investigation of non-classical correlations between observables that cannot be measured jointly. Here, I explain how the complex valued…
We consider questions related to a quantization scheme in which a classical variable f:\Omega\to R on a phase space \Omega is associated with a semispectral measure E^f, such that the moment operators of E^f are required to be of the form…
A quantum measurement, often referred to as positive operator-valued measurement (POVM), is a set of positive operators $P_j=P_j^\dag\geq 0$ summing to identity, $\sum_jP_j=\mathbb{1}$. This can be seen as a generalization of a probability…
In analogy with conventional quantum mechanics, non-commutative quantum mechanics is formulated as a quantum system on the Hilbert space of Hilbert-Schmidt operators acting on non-commutative configuration space. It is argued that the…
Based on the assumption that time evolves only in one direction and mechanical systems can be described by Lagrangeans, a dynamical C*-algebra is presented for non-relativistic particles at atomic scales. Without presupposing any…
In this article we consider means of positive bounded linear operators on a Hilbert space. We present a complete theory that provides a framework which extends the theory of the Karcher mean, its approximating matrix power means, and a…
In quantum mechanics the statistics of the outcomes of a measuring apparatus is described by a positive operator valued measure (POVM). A quantum channel transforms POVM's into POVM's, generally irreversibly, thus loosing some of the…
Descriptions of classical mechanics in Hilbert space go back to the work of Koopman and von Neumann in the 1930s. Decades later, van Hove derived a unitary representation of the group of contact transformations which recently has been used…
We offer new results and new directions in the study of operator-valued kernels and their factorizations. Our approach provides both more explicit realizations and new results, as well as new applications. These include: (i) an explicit…
We introduce positive operator-valued measure (POVM) generated by the projective unitary representation of a direct product of locally compact Abelian group $G$ with its dual $\hat G$. The method is based upon the Pontryagin duality…
Given a topological group $G$ and a unitary representation $U$ of $G$, we consider the problem of classifying the positive operator measures which are based on a $G$-homogeneous space $X$ and covariant with respect to the representation…
This paper presents an overview of close parallels that exist between the theory of positive operator-valued measures (POVMs) associated with a separable Hilbert space and the theory of frames on that space, including its most important…
We introduce the notion of a positive spectral measure on a $\sigma$-algebra, taking values in the positive projections on a Banach lattice. Such a measure generates a bounded positive representation of the bounded measurable functions. If…
In accordance with the fact that quantum measurements are described in terms of positive operator measures (POMs), we consider certain aspects of a quantization scheme in which a classical variable $f:\R^2\to \R$ is associated with a unique…
We tackle the dynamical description of the quantum measurement process, by explicitly addressing the interaction between the system under investigation with the measurement apparatus, the latter ultimately considered as macroscopic quantum…
A correspondence is established between measure-preserving, ergodic dynamics of a classical harmonic oscillator and a quantum mechanical gauge theory on two-dimensional Minkowski space. This correspondence is realized through an isometric…
Classical mechanics, in the Koopman-von Neumann formulation, is described in Hilbert space. It is shown here that classical canonical transformations are generated by Hermitian operators that are in general noncommutative. This naturally…
Observable properties of a classical physical system can be modelled deterministically as functions from the space of pure states to outcomes; dually, states can be modelled as functions from the algebra of observables to outcomes. The…
It is commonly believed that the most general type of a quantum-mechanical measurement is one described by a positive-operator valued measure (POVM). In the present paper, this statement is proven for any measurements on quantum systems…
We show that quantum measures and integrals appear naturally in any $L_2$-Hilbert space $H$. We begin by defining a decoherence operator $D(A,B)$ and it's associated $q$-measure operator $\mu (A)=D(A,A)$ on $H$. We show that these operators…