Related papers: On Classical Analogues of Free Entropy Dimension
In this thesis, I studied a mathematical development to define and quantify the uncertainty inherent in classical channels. This thesis starts with the introduction and background on how to formally think about uncertainty in the domain of…
It is well-established that quantum probability does not follow classical Kolmogorov probability calculus. Various approaches have been developed to loosen the axioms, of which the use of signed measures is the most successful (e.g. the…
We propose a new way to measure the balance between freedom and coherence in a dynamical system and a new measure of its internal variability. Based on the concept of entropy and ideas from neuroscience and information theory, we define…
We generalize the Beckner's type Poincar\'e inequality \cite{Beckner} to a large class of probability measures on an abstract Wiener space of the form $\mu\star\nu$, where $\mu$ is the reference Gaussian measure and $\nu$ is a probability…
Biane proved the free analog of the logarithmic Sobolev inequality for probability measures on the real line by means of random matrix approximation procedure. We show that the same method can be applied to reprove Biane and Voiculescu's…
Classical chaos is often characterized as exponential divergence of nearby trajectories. In many interesting cases these trajectories can be identified with geodesic curves. We define here the entropy by $S = \ln \chi (x)$ with $\chi(x)$…
We present a classical probability model appropriate to the description of quantum randomness. This tool, that we have called stochastic gauge system, constitutes a contextual scheme in which the Kolmogorov probability space depends upon…
We define and explore the classical counterpart of entanglement in complete analogy with quantum mechanics. Using a basis independent measure of entropy in the classical Hilbert space of densities that are propagated by the Frobenius-Perron…
In this paper we discuss the problem of splitting the total correlations for a bipartite quantum state described by the Von Neumann mutual information into classical and quantum parts. We propose a measure of the classical correlations as…
Borrowing the idea of topological pressure determining measure-theoretical entropy in topological dynamical systems, we establish a variational principle for upper metric mean dimension with potential in terms of upper measure-theoretical…
We give a precise functional comparison between classical and free convolutions. If $\mu$ and $\nu$ are compactly supported probability measures, we show that the expectation of $f$ over the classical convolution $\mu * \nu$ is at least the…
Quantum mechanics and classical statistical mechanics are two physical theories that share several analogies in their mathematical apparatus and physical foundations. In particular, classical statistical mechanics is hallmarked by the…
The classical stochastic model of cosmology recently developed by us is reconsidered. In that approach the parameter $w$ defined by the equation of state $p=w{\rho}$ was taken to be fluctuating with mean zero and we compared the theoretical…
We develop a numerical approach for computing the additive, multiplicative and compressive convolution operations from free probability theory. We utilize the regularity properties of free convolution to identify (pairs of) `admissible'…
We establish a connection between the relative Classical entropy and the relative Fermi-Dirac entropy, allowing to transpose, in the context of the Boltzmann or Landau equation, any entropy-entropy production inequality from one case to the…
We study algorithmic problems on subsets of Euclidean space of low fractal dimension. These spaces are the subject of intensive study in various branches of mathematics, including geometry, topology, and measure theory. There are several…
Classical probability theory supports probability measures, assigning a fixed positive real value to each event, these measures are far from satisfactory in formulating real-life occurrences. The main innovation of this paper is the…
Permutation entropy quantifies the diversity of possible orderings of the values a random or deterministic system can take, as Shannon entropy quantifies the diversity of values. We show that the metric and permutation entropy…
The correspondence principle plays a fundamental role in quantum mechanics, which naturally leads us to inquire whether it is possible to find or determine close classical analogs of quantum states in phase space -- a common meeting point…
We propose a general definition of nonequilibrium entropy of a classical stochastic field. As an example of particular interest in cosmology we apply this definition to compute the entropy of density perturbations in an inflationary…