We study decompositions of operator measures and more general sesquilinear form measures E into linear combinations of positive parts, and their diagonal vector expansions. The underlying philosophy is to represent E as a trace class valued measure of bounded variation on a new Hilbert space related to E. The choice of the auxiliary Hilbert space fixes a unique decomposition with certain proper...

It is proved that the state of maximum entropy having observed values for the n observables, X1, . . . , Xn, is the same state that minimises the matrix of covariances of any n locally unbiased estimators for n parameters for the probability distribution of X1, . . . , Xn. We sketch how to get a similar result in quantum theory, in which X1, . . . , Xn are (not necessarily commuting) quadratic ...

We consider the linear Wigner-Fokker-Planck equation subject to confining potentials which are smooth perturbations of the harmonic oscillator potential. For a certain class of perturbations we prove that the equation admits a unique stationary solution in a weighted Sobolev space. A key ingredient of the proof is a new result on the existence of spectral gaps for Fokker-Planck type operators i...

We investigate the following generalisation of the entropy of quantum measurement. Let H be an infinite-dimensional separable Hilbert space with a 'density' operator ρ, tr ρ = 1. Let I(P) ∈ R be defined for any parti-Q are physically independent). Assuming some continuity properties we give a general form of generalised information I, Theorem 1, formula (1). Throughout the paper we shall use th...

We provide a general construction of quantum generalized master equations with a memory kernel leading to well-defined, that is, completely positive and trace-preserving, time evolutions. The approach builds on an operator generalization of memory kernels appearing in the description of non-Markovian classical processes and puts into evidence the nonuniqueness of the relationship arising due to...

We introduce a separability criterion based on the positive map #:$ ̃(Tr $)!$ , where $ is a trace-class Hermitian operator. Any separable state is mapped by the tensor product of # and the identity into a nonnegative operator, which provides a simple necessary condition for separability. This condition is generally not sufficient because it is vulnerable to the dilution of entanglement. In the ...

This article generalises the concept of realised covariation to Hilbert-space-valued stochastic processes. More precisely, based on high-frequency functional data, we construct an estimator trace-class operator-valued integrated volatility process arising in general mild solutions Hilbert space-valued evolution equations sense Da Prato and Zabczyk (2014). We prove a weak law large numbers for t...