Rajesh Pereira

Differentiators and the geometry of polynomials

In 1959, Davis introduced the concept of a differentiator of an operator on a finite-dimensional Hilbert space. We prove that every such operator possesses a differentiator. We also use the theory of differentiators to solve several problems in the geometry of polynomials. For instance, we answer in the affirmative a twenty year old unsolved conjecture of Schoenberg, a related conjecture of Katsoprinakis and a fifty year old unsolved conjecture of De Bruijn and Springer.

Minimal and maximal operator spaces and operator systems in entanglement theory

Abstract We examine k -minimal and k -maximal operator spaces and operator systems, and investigate their relationships with the separability problem in quantum information theory. We show that the matrix norms that define the k -minimal operator spaces are equal to a family of norms that have been studied independently as a tool for detecting k -positive linear maps and bound entanglement. Similarly, we investigate the k -super minimal and k -super maximal operator systems that were recently introduced and show that their cones of positive elements are exactly the cones of k -block positive operators and (unnormalized) states with Schmidt number no greater than k , respectively. We characterize a class of norms on the k -super minimal operator systems and show that the completely bounded versions of these norms provide a criterion for testing the Schmidt number of a quantum state that generalizes the recently-developed separability criterion based on trace-contractive maps.

NILPOTENT MATRICES AND SPECTRALLY ARBITRARY SIGN PATTERNS

It is shown that all potentially nilpotent full sign patterns are spectrally arbitrary. A related result for sign patterns all of whose zeros lie on the main diagonal is also given. 1. Full Spectrally Arbitrary Patterns. In what follows, Mn denotes the topological vector space of all n × n matrices with real entries and Pn denotes the set of all polynomials with real coefficients of degree n or less. The superdiagonal of an n × n matrix consists of the n − 1 elements that are in the ith row and (i +1 )st column for some i ,1 ≤ i ≤ n − 1. A sign pattern is a matrix with entries in {+, 0, −} .G iven twon×n sign patterns A and B ,w e say thatB is a superpattern of A if bij = aij whenever aij � Note that a sign pattern is always a superpattern of itself. We define the function sign : R →{ +, 0, −} in the obvious way: sign(x )=+i f x> 0, sign(0) = 0, and sign(x )= − if x< 0. Given a real matrix A, sign(A) is the sign pattern with the same dimensions as A whose (i, j)th entry is sign(aij). For every sign pattern A ,w e define its associated sign pattern class to be the inverse image Q(A )= sign −1 (A). A sign pattern is said to be full if none of its entries are zero (8). A sign pattern class

Extending a characterization of majorization to infinite dimensions

Abstract We consider recent work linking majorization and trumping, two partial orders that have proven useful with respect to the entanglement transformation problem in quantum information, with general Dirichlet polynomials, Mellin transforms, and completely monotone sequences. We extend a basic majorization result to the more physically realistic infinite-dimensional setting through the use of generalized Dirichlet series and Riemann–Stieltjes integrals.

Spherical designs and anticoherent spin states

Anticoherent spin states are quantum states that exhibit maximally nonclassical behaviour in a certain sense. Any spin state whose Majorana representation is a Platonic solid is called a perfect state. By direct calculation, it has been shown that any perfect state is an anticoherent spin state. We show that any spin state whose Majorana representation is both the orbit of a finite subgroup of O(3) and a spherical t-design must be an anticoherent spin state of order t. Since all Platonic solids are spherical designs, this result gives an explanation of the anticoherence of perfect states and explains their observed order. We also show that any spin state whose Majorana representation lies in any single open hemisphere cannot be anticoherent of any order. This result is then used to give further relations between spherical designs and anticoherent spin states. We also pose some questions relating spherical designs and geometric entanglement.

Representing conditional expectations as elementary operators

Let A be a C*-algebra and let B be a C*-subalgebra of A. We call a linear operator from A to B an elementary conditional expectation if it is simultaneously an elementary operator and a conditional expectation of A onto B. We give necessary and sufficient conditions for the existence of a faithful elementary conditional expectation of a prime unital C*-algebra onto a subalgebra containing the identity element. We give a description of all faithful elementary conditional expectations. We then use these results to give necessary and sufficient conditions for certain conditional expectations to be index-finite (in the sense of Watatani) and we derive an inequality for the index.

Orderings on semirings and completely positive matrices

A real symmetric matrix is called a completely positive matrix if there exists a nonnegative real matrix such that . In this paper, we extend the notion of complete positivity for matrices over real numbers to matrices over semirings in general. We find necessary and sufficient conditions for matrices over certain semirings to be completely positive. We also find an upper bound on the CP-rank of completely positive matrices over certain special types of semirings.

Dirichlet polynomials, majorization, and trumping

Majorization and trumping are two partial orders which have proved useful in quantum information theory. We show some relations between these two partial orders and generalized Dirichlet polynomials, Mellin transforms, and completely monotone functions. These relations are used to prove a succinct generalization of Turgut’s characterization of trumping.

Borcea’s Variance Conjectures on the Critical Points of Polynomials

Closely following recent ideas of J. Borcea, we discuss various modifications and relaxations of Sendov’s conjecture about the location of critical points of a polynomial with complex coefficients. The resulting open problems are formulated in terms of matrix theory, mathematical statistics or potential theory. Quite a few links between classical works in the geometry of polynomials and recent advances in the location of spectra of small rank perturbations of structured matrices are established. A couple of simple examples provide natural and sometimes sharp bounds for the proposed conjectures.

Operator structures and quantum one-way LOCC conditions

We conduct the first detailed analysis in quantum information of recently derived operator relations from the study of quantum one-way local operations and classical communications (LOCC). We show how operator structures such as operator systems, operator algebras, and Hilbert C*-modules naturally arise in this setting. We make use of these structures to derive new results and bounds in the study of one-way LOCC, and we use the approach to uncover new derivations of some previously established results.