We discuss the construction of relational observables in time-reparametrization invariant quantum mechanics and we argue that their physical interpretation can be understood terms conditional probabilities, which are defined from solutions constraint equation a generalization Page-Wootters formalism. In this regard, show how expectation values worldline tensor fields related to averages suitabl...

Let A be an m n (m n) complex matrix. It is known that there is a unique polar decomposition A = QH, where Q Q = I, the n n identity matrix, and H is positive de nite, provided A has full column rank. This note addresses the following question: how much may Q change if A is perturbed? For the square case m = n our bound, which is valid for any unitarily invariant norm, is sharper and simpler th...

The separability modulus l(ρ) of a state ρ of an arbitrary finite composite quantum system, is the largest t in [0, 1] such that t.ρ+(1− t).τ is unentangled, where τ is the normalized trace. The basic properties of l, introduced by Vidal & Tarrach [1] in another guise, are briefly established. With these properties, we obtain conditions on the spectrum of a state which imply that it is separabl...

For a large class of unitarily invariant reproducing kernel functions K on the unit ball \(\mathbb B_d\) in C^d\), we characterize K-inner as admitting suitable transfer function realization. We associate with each K-contraction T ∈ L(H)d canonical operator-valued and extend uniqueness theorem Arveson for minimal K-dilations to our setting. thus generalize results Olofsson m-hypercontractions d...

We point out a simple criterion for convergence of polynomials to concrete entire function in the Laguerre-Pólya ( LP ) class (of all functions arising as uniform limits with only real roots). then use this show that any random can be obtained limit rescaled characteristic principal submatrices an infinite unitarily invariant Hermitian matrix. Conversely, matrix converge uniformly function. Thi...

We give an overview over the higher torsion invariants of Bismut-Lott, Igusa-Klein and Dwyer-Weiss-Williams, including some more or less recent developments. The classical Franz-Reidemeister torsion τFR is an invariant of manifolds with acyclic unitarily flat vector bundles [62], [33]. In contrast to most other algebraic-topological invariants known at that time, it is invariant under homeomorp...