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We explore the set of unitary matrices characterized by a given structure in the context of their applications in the field of Quantum Information. In the first part of the Thesis we focus on classification of special classes of unitary…
This paper is devoted to the study of the separability problem in the field of Quantum information theory. We deal mainly with the bipartite finite dimensional case and with two types of matrices, one of them being the PPT matrices. We…
Based on maximally entangled states, we explore the constructions of mutually unbiased bases in bipartite quantum systems. We present a new way to construct mutually unbiased bases by difference matrices in the theory of combinatorial…
We study $n$-dimensional matrices with $\{0,1\}$-entries ($n$-cubes) such that all their $2$-dimensional slices are incidence matrices of symmetric designs. A known construction of these objects obtained from difference sets is generalized…
This paper introduces the concept of a generating set for stochastic matrices -- a subset of matrices whose repeated composition generates the entire set. Understanding such generating sets requires specifying the "indivisible elements" and…
This work introduces a new class of symmetric matrix structures, called harmonic structures, which enable the generation of all possible directed transitions $(x_i, x_{i+1})$ over a set of $n$ symbols, without internal repetitions. Unlike…
We reduce the question whether a given quantum mixed state is separable or entangled to the problem of existence of a certain full family of commuting normal matrices whose matrix elements are partially determined by components of the pure…
Superregular matrices, i.e., matrices where all square submatrices are non-singular, have a wide range of applications in communications. A superregular block matrix is a broader concept where all full block submatrices, with the…
We introduce a new family of indecomposable positive linear maps based on entangled quantum states. Central to our construction is the notion of an unextendible product basis. The construction lets us create indecomposable positive linear…
Hadamard matrices in $\{0,1\}$ presentation are square $m\times m$ matrices whose entries are zeros and ones and whose rows considered as vectors in $\Bbb R^m$ produce the Gram matrix of a special form with respect to the standard scalar…
We develop an extension of the process matrix (PM) framework for correlations between quantum operations with no causal order that allows multiple rounds of information exchange for each party compatibly with the assumption of well-defined…
The construction of large-scale quantum computers will require modular architectures that allow physical resources to be localized in easy-to-manage packages. In this work, we examine the impact of different graph structures on the…
Conference matrices are used to define complex structures on real vector spaces. Certain lattices in these spaces become modules for rings of quadratic integers. Multiplication of these lattices by non-principal ideals yields simple…
The decomposition of large unitary matrices into smaller ones is important, because it provides ways to realization of classical and quantum information processing schemes. Today, most of the methods use planar meshes of tunable two-channel…
In science and engineering, intelligent processing of complex signals such as images, sound or language is often performed by a parameterized hierarchy of nonlinear processing layers, sometimes biologically inspired. Hierarchical systems…
Multiplication of a sparse matrix with another (dense or sparse) matrix is a fundamental operation that captures the computational patterns of many data science applications, including but not limited to graph algorithms, sparsely connected…
We extend the construction of so-called encapsulated global summation-by-parts operators to the general case of a mesh which is not boundary conforming. Owing to this development, energy stable discretizations of nonlinear and variable…
We give necessary and sufficient conditions under which a density matrix acting on a two-fold tensor product space is separable. Our conditions are given in terms of quantum conditional information transmission.
In computational design and fabrication, neural networks are becoming important surrogates for bulky forward simulations. A long-standing, intertwined question is that of inverse design: how to compute a design that satisfies a desired…
While the standard formulation of quantum theory assumes a fixed background causal structure, one can relax this assumption within the so-called process matrix framework. Remarkably, some processes, termed causally nonseparable, are…