Related papers: Distinguishability notion based on Wootters statis…

By assigning a probability measure via the spectrum of the normalized Laplacian to each graph and using L^p Wasserstein distances between probability measures, we define the corresponding spectral distances d_p on the set of all graphs.…

We introduce a novel formulation for geometry on discrete points. It is based on a universal differential calculus, which gives a geometric description of a discrete set by the algebra of functions. We expand this mathematical framework so…

A finite set of unlabelled points in Euclidean space is the simplest representation of many real objects from mineral rocks to sculptures. Since most solid objects are rigid, their natural equivalence is rigid motion or isometry maintaining…

Probability distributions play a central role in quantum mechanics, and even more so in quantum optics with its rich diversity of theoretically conceivable and experimentally accessible quantum states of light. Quantifiers that compare two…

We analyse the problem of finding sets of quantum states that can be deterministically discriminated. From a geometric point of view this problem is equivalent to that of embedding a simplex of points whose distances are maximal with…

A group of symmetric operators are introduced to carry out the separability criterion for bipartite and multipartite quantum states. Every symmetric operator, represented by a symmetric matrix with only two nonzero elements, and their…

The principle of local distinguishability states that an arbitrary physical state of a bipartite system can be determined by the combined statistics of local measurements performed on the subsystems. A necessary and sufficient requirement…

We introduce a new technique to detect separable states using semidefinite programs. This approach provides a sufficient condition for separability of a state that is based on the existence of a certain local linear map applied to a known…

A periodic perturbation generates a complicated dynamics close to separatrices and saddle points. We construct an asymptotic solution which is close to the separatrix for the unperturbed Duffing's oscillator over a long time. This solution…

Random geometric graphs are a popular choice for a latent points generative model for networks. Their definition is based on a sample of $n$ points $X_1,X_2,\cdots,X_n$ on the Euclidean sphere~$\mathbb{S}^{d-1}$ which represents the latent…

A network is called localizable if the positions of all the nodes of the network can be computed uniquely. If a network is localizable and embedded in plane with generic configuration, the positions of the nodes may be computed uniquely in…

Metrics on the space of sets of trajectories are important for scientists in the field of computer vision, machine learning, robotics, and general artificial intelligence. However, existing notions of closeness between sets of trajectories…

We study discretizations of Hamiltonian systems on the probability density manifold equipped with the $L^2$-Wasserstein metric. Based on discrete optimal transport theory, several Hamiltonian systems on graph (lattice) with different…

The paper is devoted to a categorical study of the category of probabilistic metric spaces. The study is based on an isomorphic description of the category of probabilistic metric spaces. The isomorphic description was obtained in [3] and…

We propose a novel measure of statistical depth, the metric spatial depth, for data residing in an arbitrary metric space. The measure assigns high (low) values for points located near (far away from) the bulk of the data distribution,…

One-dimensional discrete-time quantum walks show a rich spectrum of topological phases that have so far been exclusively analysed in momentum space. In this work we introduce an alternative approach to topology which is based on the…

We study rare events in systems of diffusive fields driven out of equilibrium by the boundaries. We present a numerical technique and use it to calculate the probabilities of rare events in one and two dimensions. Using this technique, we…

We prove strong statistical stability of a large class of one-dimensional maps which may have an arbitrary finite number of discontinuities and of non-degenerate critical points and/or singular points with infinite derivative, and satisfy…

In spaces of metrics, we investigate topological distributions of the doubling property, the uniform disconnectedness, and the uniform perfectness, which are the quasi-symmetrically invariant properties appearing in the David--Semmes…

We experimentally demonstrate that the statistical properties of distances between pedestrians which are hindered from avoiding each other are described by the Gaussian Unitary Ensemble of random matrices. The same result has recently been…