Related papers: Maximal inequalities in quantum probability spaces

Let $r$ and $n$ be positive integers such that $r<n$, and $\mathbb{K}$ be an arbitrary field. In a recent work, we have determined the maximal dimension for a linear subspace of $n$ by $n$ symmetric matrices with rank less than or equal to…

In this paper, we consider the predual spaces of weak Orlicz spaces. As an application, we provide the Fefferman-Stein vector-valued maximal inequality for the weak Orlicz spaces. In order to prove this statement, we introduced the…

We define a covariance-type operator on Wiener space: for F and G two random variables in the Gross-Sobolev space $D^{1,2}$ of random variables with a square-integrable Malliavin derivative, we let $Gamma_{F,G}=$ where $D$ is the Malliavin…

Review of tomographic probability representation of quantum states is presented both for oscillator systems with continious variables and spin--systems with discrete variables. New entropic--information inequalities are obtained for…

We introduce a generalized framework for private quantum codes using von Neumann algebras and the structure of commutants. This leads naturally to a more general notion of complementary channel, which we use to establish a generalized…

We review the plethora of uncertainty relations that appear in quantum mechanics and their nuances. We present both foundational applications, e.g. in understanding and defining complementarity, and practical applications, e.g. in quantum…

Quantum inequalities bound the extent to which weighted time averages of the renormalized energy density of a quantum field can be negative. They have mostly been proved in flat spacetime, but we need curved-spacetime inequalities to…

Attention is focused on antisymmetrized versions of quantum spaces that are of particular importance in physics, i.e. two-dimensional quantum plane, q-deformed Euclidean space in three or four dimensions as well as q-deformed Minkowski…

We explore the uncertainty relation for unitary operators in a new way and find two uncertainty equalities for unitary operators, which are minimized by any pure states. Additionally, we derive two sets of uncertainty inequalities that…

The derivation of Bell inequalities in terms of quantum statistical (thermodynamic) entropies is considered. Inequalities of the Wigner form are derived but shown to be extremely limiting in their applicability due to the nature of the…

In this paper we set up a general formalism to deal with quantum theories on a Lobatchevski space, i.e. a spatial manifold that is homogeneous, isotropic and has negative curvature. The heart of our approach is the construction of a…

We establish a weighted inequality for fractional maximal and convolution type operators, between weak Lebesgue spaces and Wiener amalgam type spaces on $ \mathbb R $ endowed with a measure which needs not to be doubling.

We prove a sharp Sobolev inequality on manifolds with nonnegative Ricci curvature. Moreover, we prove a Michael-Simon inequality for submanifolds in manifolds with nonnegative sectional curvature. Both inequalities depend on the asymptotic…

Sharp uncertainty relations restricting the values of variances in the position space and in the momentum (wavevector) space are derived. They have the same form $\Delta r\Delta k\ge 5/2$ in the classical theory of light beams, in the…

We extend the Paley--Wiener theorem for riemannian symmetric spaces to an important class of infinite dimensional symmetric spaces. For this we define a notion of propagation of symmetric spaces and examine the direct (injective) limit…

Many statistical estimation procedures lead to nonconvex optimization problems. Algorithms to solve these are often guaranteed to output a stationary point of the optimization problem. Oracle inequalities are an important theoretical…

Certain many-particle Hardy inequalities are derived in a simple and systematic way using the so-called ground state representation for the Laplacian on a subdomain of $\mathbb{R}^n$. This includes geometric extensions of the standard Hardy…

Covariant classical particle dynamics is described, and the associated covariant relativistic particle quantum mechanics is derived. The invariant symmetric bracket is defined on the space of quantum amplitudes, and its relation to a…

We study coercive inequalities in Orlicz spaces associated to the probability measures on finite and infinite dimensional spaces which tails decay slower than the Gaussian ones. We provide necessary and sufficient criteria for such…

Krein space quantization and the ambient space formalism have been successfully applied to address challenges in quantum geometry (e.g., quantum gravity) and the axiomatic formulation of quantum Yang-Mills theory, including phenomena such…