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A particularly simple description of separability of quantum states arises naturally in the setting of complex algebraic geometry, via the Segre embedding. This is a map describing how to take products of projective Hilbert spaces. In this…
In this paper I will investigate geometrical structures of multipartite quantum systems based on complex projective varieties. These varieties are important in characterization of quantum entangled states. In particular I will establish…
Quantum Entanglement is one of the key manifestations of quantum mechanics that separate the quantum realm from the classical one. Characterization of entanglement as a physical resource for quantum technology became of uppermost…
We study the noncommutative geometrical structures of quantum entangled states. We show that the space of a pure entangled state is a noncommutative space. In particular we show that by rewritten the conifold or the Segre variety we can get…
The past few years have seen a revived interest in quantum geometrical characterizations of band structures due to the rapid development of topological insulators and semi-metals. Although the metric tensor has been connected to many…
This article is an expository account aimed at viewing entanglement in finite-dimensional quantum many-body systems as a phenomenon of global geometry. While the mathematics of general quantum states has been studied extensively, this…
Segre embedding was introduced by C. Segre (1863--1924) in his famous 1891 article \cite{segre}. The Segre embedding plays an important roles in algebraic geometry as well as in differential geometry, mathematical physics, and coding…
We show that the secant variety of the Segre variety gives useful information about the geometrical structure of an arbitrary multipartite quantum system. In particular, we investigate the relation between arbitrary bipartite and…
We establish a relation between topological and quantum entanglement for a multi-qubit state by considering the unitary representations of the Artin braid group. We construct topological operators that can entangle multi-qubit state. In…
We investigate a recent conjecture connecting the AdS/CFT correspondence and entanglement renormalization tensor network states (MERA). The proposal interprets the tensor connectivity of the MERA states associated to quantum many body…
We study the quantum entanglement structure of integer quantum Hall states via the reduced density matrix of spatial subregions. In particular, we examine the eigenstates, spectrum and entanglement entropy (EE) of the density matrix for…
A well-known noncommutative deformation $\mathcal A^N_{\mathbf{q}}$ of the polynomial algebra $\mathcal A^N$ can be obtained as a twist of $\mathcal A^N$ by a cocycle on the grading semigroup. Of particular interest to us is an…
In quantum physics, multiparticle systems are described by quantum states acting on tensor products of Hilbert spaces. This product structure leads to the distinction between product states and entangled states; moreover, one can quantify…
We establish relations between Segre variety, conifold, Hopf fibration, and separable sets of pure two-qubit states. Moreover, we investigate the geometry and topology of separable sets of pure multi-qubit states based on a complex…
One of the most important questions in quantum information theory is the so-called separability problem. It involves characterizing the set of separable (or, equivalently entangled) states among mixed states of a multipartite quantum…
Segre classes encode essential intersection-theoretic information concerning vector bundles and embeddings of schemes. In this paper we survey a range of applications of Segre classes to the definition and study of invariants of singular…
We propose a new approach to the geometry of the four-qubit entanglement classes depending on parameters. More precisely, we use invariant theory and algebraic geometry to describe various stratifications of the Hilbert space by SLOCC…
We express the Segre class of a monomial scheme -- or, more generally, a scheme monomially supported on a set of divisors cutting out complete intersections -- in terms of an integral computed over an associated body in euclidean space. The…
We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i.) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii.)…
The aim of the paper is to propose geometric descriptions of multipartite entangled states using algebraic geometry. In the context of this paper, geometric means each stratum of the Hilbert space, corresponding to an entangled state, is an…