Related papers: On Classical Analogues of Free Entropy Dimension

There is a well-known mathematical similarity between two-dimensional classical polarization optics and two-level quantum systems, where the Poincare and Bloch spheres are identical mathematical structures. This analogy implies that the…

We propose a new definition of preimage entropy dimension for continuous maps on compact metric spaces, investigate fundamental properties of the preimage entropy dimension, and compare the preimage entropy dimension with the topological…

We derive exceedingly simple practical procedures revealing the quantum nature of states and measurements by the violation of classical upper bounds on the statistics of arbitrary measurements. Data analysis is minimum and definite…

Under the assumption of a natural subadditive potential, the so called cylinder function, working on the symbol space we prove the existence of the ergodic invariant probability measure satisfying the equilibrium state. As an application we…

Both classical and respectively quantum observables can be modeled as somewhat similar examples of random variables. In such a model the associated measurements preserve the values spectrum of an observable but change the corresponding…

The concept of typicality refers to properties holding for the "overwhelming majority" of cases and is a fundamental idea of the qualitative approach to dynamical problems. We argue that measure-theoretical typicality would be the adequate…

We define a one-parameter family of entropies, each assigning a real number to any probability measure on a compact metric space (or, more generally, a compact Hausdorff space with a notion of similarity between points). These entropies…

We introduce and study a new type of convolution of probability measures called the orthogonal convolution, which is related to the monotone convolution. Using this convolution, we derive alternating decompositions of the free additive…

Black hole entropy appears to be ``universal''--many independent calculations, involving models with very different microscopic degrees of freedom, all yield the same density of states. I discuss the proposal that this universality comes…

Cosmological fine-tuning has traditionally been associated with the narrowness of the intervals in which the parameters of the physical models must be located to make life possible. A more thorough approach focuses on the probability of the…

Entanglement entropy in causal sets offers a fundamentally covariant characterisation of quantum field degrees of freedom. A known result in this context is that the degrees of freedom consist of a number of contributions that have…

The aim of this paper is to show that a probability measure concentrates independently of the dimension like a gaussian measure if and only if it verifies Talagrand's $\T_2$ transportation-cost inequality. This theorem permits us to give a…

There is a class of higher derivative gravity theories that are in some sense natural extensions of cosmological Einstein's gravity with a unique maximally symmetric classical vacuum and only a massless spin-2 excitation about the vacuum…

We report an inconsistency found in probability theory (also referred to as measure-theoretic probability). For probability measures induced by real-valued random variables, we deduce an "equality" such that one side of the "equality" is a…

In statistical mechanics, measuring the number of available states and their probabilities, and thus the system's entropy, enables the prediction of the macroscopic properties of a physical system at equilibrium. This predictive capacity…

We prove that a probability measure on an abstract metric space satisfies a non trivial dimension free concentration inequality for the $\ell_2$ metric if and only if it satisfies the Poincar\'e inequality.

We study various measure theories using the classical approach and then compute the Hausdorff dimension of some simple objects and self-similar fractals. We then develop a nonstandard approach to these measure theories and examine the…

Quantum addition channels have been recently introduced in the context of deriving entropic power inequalities for finite dimensional quantum systems. We prove a reverse entropy power equality which can be used to analytically prove an…

It has long been conjectured that the entropy of quantum fields across boundaries scales as the boundary area. This conjecture has not been easy to test in spacetime dimensions greater than four because of divergences in the von Neumann…

Motivated by Voiculescu's liberation theory, we introduce the orbital free entropy $\chi_orb$ for non-commutative self-adjoint random variables (also for "hyperfinite random multi-variables"). Besides its basic properties the relation of…