Related papers: A Kolmogorov Extension Theorem for POVMs

Let $P_1,\dots,P_m\in\mathbb{Z}[y]$ be any linearly independent polynomials with zero constant term. We show that there exists a $\gamma>0$ such that any subset of $\mathbb{F}_q$ of size at least $q^{1-\gamma}$ contains a nontrivial…

We discuss Lomonosov's proof of the Pontryagin-Krein Theorem on invariant maximal non-positive subspaces, prove the refinement of one theorem from \cite{OShT} on common fixed points for a group of fractional-linear maps of operator ball and…

The modal interpretation of quantum mechanics allows one to keep the standard classical definition of realism intact. That is, variables have a definite status for all time and a measurement only tells us which value it had. However, at…

We establish a number of "concatenation theorems" that assert, roughly speaking, that if a function exhibits "polynomial" (or "Gowers anti-uniform", "uniformly almost periodic", or "nilsequence") behaviour in two different directions…

The Parameter Continuation Theorem is the theoretical foundation for polynomial homotopy continuation, which is one of the main tools in computational algebraic geometry. In this note, we give a short proof using Gr\"obner bases. Our…

Although time measurements are routinely performed in laboratories, their theoretical description is still an open problem. Correspondingly, the status of the energy-time uncertainty relation is unsettled. In the first part of this work the…

Here we propose an implementation of all possible Positive Operator Value Measures (POVMs) of two-photon polarization states. POVMs are the most general class of quantum measurements. Our setup requires linear optics, Bell State…

We prove a superposition principle for nonlinear Fokker-Planck-Kolmogorov equations on Euclidean spaces and their corresponding linearized first-order continuity equation over the space of Borel (sub-)probability measures. As a consequence,…

Given a sequence of orthogonal polynomials $(p_n)_n$ with respect to a positive measure in the real line, we study the real zeros of finite combinations of $K+1$ consecutive orthogonal polynomials of the form $$…

We obtain asymptotic expansions for probabilities $\mathbb{P}(S_N=k)$ of partial sums of uniformly bounded integer-valued functionals $S_N=\sum_{n=1}^N f_n(X_n)$ of uniformly elliptic inhomogeneous Markov chains. The expansions involve…

The famous Lomonosov's invariant subspace theorem states that if a continuous linear operator T on an infinite-dimensional normed space E "commutes" with a compact nonzero operator K, i.e., TK=KT, then T has a non-trivial closed invariant…

The quantum-mechanical rule for probabilities, in its most general form of positive-operator valued measure (POVM), is shown to be a consequence of the environment-assisted invariance (envariance) idea suggested by Zurek [Phys. Rev. Lett.…

The expansion exponent (or expansion constant) for maps was introduced by Schreiber in \cite{s}. In this paper, we introduce the analogous exponent for measures. We shall prove the following results: The expansion exponent of a measurable…

We establish a direct connection between two fundamental topics: one in probability theory and one in quantum field theory. The first topic is the problem of pointwise multiplication of random Schwartz distributions which has been the…

We revisit the $R-$positivity of nearest neighbors matrices on ${\ZZ_+}$ and the Gibbs measures on the set of nearest neighbors trajectories on ${\ZZ_+}$ whose Hamiltonians award either visits to sites a or visits to edges. We give…

Variable Length Memory Chains (VLMC), which are generalizations of finite order Markov chains, turn out to be an essential tool to modelize random sequences in many domains, as well as an interesting object in contemporary probability…

The celebrated theorem of Gromov asserts that any finitely generated group with polynomial growth contains a nilpotent subgroup of finite index. Alternative proofs have been given by Kleiner and others. In this note, we give yet another…

We define a complete measurement of a quantum observable (POVM) as a measurement of the maximally refined version of the POVM. Complete measurements give information from the multiplicities of the measurement outcomes and can be viewed as…

Let $d\mu$ be a probability measure on the unit circle and $d\nu$ be the measure formed by adding a pure point to $d\mu$. We give a simple formula for the Verblunsky coefficients of $d\nu$ based on a result of Simon. Then we consider…

Let $G$ be a finitely generated group equipped with a symmetric generating $% k $-tuple $S$. Let $|\cdot|$ and $V$ be the associated word length and volume growth function. Let $\nu$ be a probability measure such that $% \nu(g)\simeq…