Related papers: Decomposable Pauli diagonal maps and Tensor Square…

Let $S$ be a Riemann surface of type $(p,n)$ with $3p-3+n>0$. Let $\omega$ be a pseudo-Anosov map of $S$ that is obtained from Dehn twists along two families $\{A,B\}$ of simple closed geodesics that fill $S$. Then $\omega$ can be realized…

We answer a question of Schleicher by showing that, for an exponential map with nonescaping singular value, every periodic ray lands. This is an analog of a theorem of Douady and Hubbard concerning polynomials. We also prove a partial…

We extend the theory of equitable decompositions, in which, if a graph has a particular type of symmetry, i.e. a uniform or basic automorphism $\phi$, it is possible to use $\phi$ to decompose a matrix $M$ appropriately associated with the…

We study mapping cones and their dual cones of positive maps of the n by n matrices into itself. For a natural class of cones there is a close relationship between maps in the cone, super-positive maps, and separable states. In particular…

Let M be a matrix whose entries are power series in several variables and determinant det(M) does not vanish identically. The equation det(M)=0 defines a hypersurface singularity and the (co)-kernel of M is a maximally Cohen-Macaulay module…

We study the diagonal mappings in non-involutive set-theoretic solutions of the Yang-Baxter equation. We show that, for non-degenerate solutions, they are commuting bijections. This gives the positive answer to the question: ``Is every…

Unitary decomposition is a widely used method to map quantum algorithms to an arbitrary set of quantum gates. Efficient implementation of this decomposition allows for translation of bigger unitary gates into elementary quantum operations,…

Multi-controlled Pauli gates are typical high-level qubit operations that appear in the quantum circuits of various quantum algorithms. We find multi-controlled Pauli gate decompositions with smaller CNOT-count or $T$-depth while keeping…

We analyze a class of piecewise linear parabolic maps on the torus, namely those obtained by considering a linear map with double eigenvalue one and taking modulo one in each component. We show that within this two parameter family of maps,…

An $n\times n$ symmetric matrix $A$ is copositive if the quadratic form $x^TAx$ is nonnegative on the nonnegative orthant $\mathbb{R}^{n}_{\geq 0}$. The cone of copositive matrices contains the cone of matrices which are the sum of a…

Any Hilbert space with composite dimension can be factorized into a tensor product of smaller Hilbert spaces. This allows to decompose a quantum system into subsystems. We propose a simple tractable model for a constructive study of…

In this paper, we investigate the general properties and structure of $C^*$-extreme points within the $C^*$-convex set $\mathrm{UCP}(\mathcal{A},B(\mathcal{H}))$ of all unital completely positive (UCP) maps from a unital real $C^*$-algebra…

We classify those rational maps $f: \mathbb{P}^1 \to \mathbb{P}^1$ for which there exists a contravariant tensor $q$ which is parallel, i.e. such that $f^*q // q$, by proving that such maps preserve a parabolic orbifold.

Let $X$ be a complex submanifold of dimension $d$ of $\mathbb P^m\times\mathbb P^n$ ($m\geq n\geq 2$) and denote by $\alpha\colon\Pic(\mathbb P^m\times\mathbb P^n)\to \Pic(X)$ the restriction map of Picard groups, by $N_{X|\mathbb…

We prove that any compact surface with constant positive curvature and conical singularities can be decomposed into irreducible components of standard shape, glued along geodesic arcs connecting conical singularities. This is a spherical…

We show that the covariance matrix of a quantum state can be reconstructed from position measurements using the simple notion of polar duality, familiar from convex geometry. In particular, all multidimensional Gaussian states (pure or…

We study the problem when every matrix over a division ring is representable as either the product of traceless matrices or the product of semi-traceless matrices, and also give some applications of such decompositions. Specifically, we…

We prove that a positive proportion of integers are expressible as the sum of two rational cubes, and a positive proportion are not so expressible, thus proving a conjecture of Davenport. More generally, we prove that a positive proportion…

We show that the set of not-completely-positive (NCP) maps is unbounded, unless further assumptions are made. This is done by first proposing a reasonable definition of a valid NCP map, which is nontrivial because NCP maps may lack a full…

Motivated by questions from Ehrhart theory, we present new results on discrete equidecomposability. Two rational polygons $P$ and $Q$ are said to be discretely equidecomposable if there exists a piecewise affine-unimodular bijection…