Related papers: Towards a Universal Measure of Complexity
Complexity is an important metric for appropriate characterization of different classes of irregular signals, observed in the laboratory or in nature. The literature is already rich in the description of such measures using a variety of…
Complexity theory embodies some of the hardest, most fundamental and most challenging open problems in modern science. The very term complexity is very elusive, so that the main goal of this theory is to find meaningful quantifiers for it.…
We extend algorithmic information theory to quantum mechanics, taking a universal semicomputable density matrix (``universal probability'') as a starting point, and define complexity (an operator) as its negative logarithm. A number of…
The fundamental impasses and ruptures in various domains of the canonical, unitary science, or the 'end of science', become the more and more evident. The natural unity of being is recovered within a universal nonperturbative method leading…
An accurate assessment of a model's complexity is crucial for topics such as interpretation, generalization, and model selection. However, most existing complexity measures either rely on heuristic assumptions or are computationally…
Compact data representations are one approach for improving generalization of learned functions. We explicitly illustrate the relationship between entropy and cardinality, both measures of compactness, including how gradient descent on the…
We show that probabilities of results of all possible measurements performing on a quantum system depend on the system's state only through its density matrix. Therefore all experimentally available information about the state contains in…
We investigate the relationship between mixedness and entanglement for Gaussian states of continuous variable systems. We introduce generalized entropies based on Schatten $p$-norms to quantify the mixedness of a state, and derive their…
We study some measures which are related to the notion of the $\e$-complexity. We prove that measure of $\e$-complexity defined on the base of the notion of $\e$-separability is equivalent to the dual measure that is defined through…
There are at least a number of ways to formally define complexity. Most of them relate to some kind of minimal description of the studied object. Being this one in form of minimal resources of minimal effort needed to generate the object…
It is well known that for pure states the relative entropy of entanglement is equal to the reduced entropy, and the closest separable state is explicitly known as well. The same holds for Renyi relative entropy per recent results. We ask…
In the paper we present results to develop an irreducible theory of complex systems in terms of self-organization processes of prime integer relations. Based on the integers and controlled by arithmetic only the self-organization processes…
In the last three decades, several measures of complexity have been proposed. Up to this point, most of such measures have only been developed for finite spaces. In these scenarios the baseline distribution is uniform. This makes sense…
We observe an infinite sequence of independent identically distributed random variables $X_1,X_2,\ldots$ drawn from an unknown distribution $p$ over $[n]$, and our goal is to estimate the entropy $H(p)=-\mathbb{E}[\log p(X)]$ within an…
We consider entropy in Generalized Non-Signalling Theory (also known as box world) where the most common definition of entropy is the measurement entropy. In this setting, we completely characterize the set of allowed entropies for a…
Measurement incompatibility is one of the basic aspects of quantum theory. Here we study the structure of the set of compatible -- i.e. jointly measurable -- measurements. We are interested in whether or not there exist compatible…
A simple entanglement measure for multipartite pure states is formulated based on the partial entropy of a series of reduced density matrices. Use of the proposed new measure to distinguish disentangled, partially entangled, and maximally…
In estimating the complexity of objects, in particular of graphs, it is common practice to rely on graph- and information-theoretic measures. Here, using integer sequences with properties such as Borel normality, we explain how these…
We study finitely additive extensions of the asymptotic density to all the subsets of natural numbers. Such measures are called density measures. We consider a class of density measures constructed from free ultrafilters on $\mathbb{N}$ and…
Assessing quantitatively the state and dynamics of a social system is a very difficult problem. It is of great importance for both practical and theoretical reasons such as establishing the efficiency of social action programs, detecting…