In this seminar report, we present in detail the proof of a recent result due to J. Briët and T. Vidick, improving an estimate in a 2008 paper by D. Pérez-Garćıa, M. Wolf, C. Palazuelos, I. Villanueva, and M. Junge, estimating the growth of the deviation in the tripartite Bell inequality. The proof requires a delicate estimate of the norms of certain trilinear (or d-linear) forms on Hilbert spa...

We examine Hilbert-Schmidt stability (HS-stability) of discrete amenable groups from several angles. give a short, elementary proof that finitely generated nilpotent are HS-stable. investigate the permanence HS-stability under central quotients by showing is preserved finite quotients, but not in general. characterization for semidirect products G⋊γZ with G abelian. use it to construct first ex...

Let A be a Banach algebra with a Hilbert space norm (norm defined by a scalar product). We shall call A a right complemented algebra if it has the property that the orthogonal complement of a right ideal is again a right ideal. This notion was introduced in the author's doctoral thesis [5]. It was proved that under certain additional assumptions every right complemented algebra is left compleme...

In this paper we show that the output of a nonlinear system with inputs in (L2 [0, T]; R ' ) whose state satisfies a nonlinear differential equation with standard smoothness conditions can be written as the composition of a nonlinear map with a linear Hilbert-Schmidt operator acting on the input. The result also extends to abstract semi-linear infinite dimensional systems. The approach is via t...

In this paper, we prove that each of the following functions is convex on R: f(t) = wN(AtXA1?t ? A1?tXAt), g(t) wN(AtXA1?t), and h(t) wN(AtXAt) where A > 0, X Mn N(.) a unitarily invariant norm onMn. Consequently, answer positively question concerning convexity function t w(AtXAt) proposed by in (2018). We provide some generalizations extensions wN(.) using Kwong functions. More precisely, w...

In this article we study convexity properties of distance functions in infinite dimensional Finsler unitary groups, such as the full group, Schatten perturbations identity and groups finite von Neumann algebras. The structures are defined by translation different norms on tangent space at identity. We first prove a result for metric derived from operator norm group. also strong results squared ...

Entanglement is defined for each vector subspace of the tensor product of two finite-dimensional Hilbert spaces, by applying the notion of operator entanglement to the projection operator onto that subspace. The operator Schmidt decomposition of the projection operator defines a string of Schmidt coefficients for each subspace, and this string is assumed to characterize its entanglement, so tha...

Let ? be an open connected subset of the complex plane C and let T be a bounded linear operator on a Hilbert space H. For ? in ? let e the orthogonal projection onto the null-space of T-?I . We discuss the necessary and sufficient conditions for the map ?? to b e continuous on ?. A generalized Gram- Schmidt process is also given.

By considering an exactly solvable model of a two interacting spin-1/2 qubits described by the Heisenberg anisotropic interaction in presence intrinsic decoherence, we study dynamics entanglement quantified concurrence and measurement-induced nonlocality (MIN) based on Hilbert-Schmidt norm trace distance with different initial conditions. We highlight relationship between MIN for pure state. Fo...