Combinatorics and Algebra Seminar-Quantum informatics meets bioinformatics: combinatorial enumeration of special breakpoints graphs with application to calculating entanglement

Title: Quantum informatics meets bioinformatics: combinatorial enumeration of special breakpoints graphs with application to calculating entanglement

Speaker: Max Alekseyev, GWU
Date and time: Monday, January 30, 4–5 pm
Place: Rome 206

Abstract: Breakpoint graphs were first introduced for the analysis of genome evolution almost three decades ago, and since then they have been actively studied and used in a variety of bioinformatics applications.
In the simplest form breakpoint graphs represent edge-colored (multii)graphs formed by 3 perfect matchings of different colors. Every pair of colors in a breakpoint graph defines a collection of cycles, in which the colors of edges alternate. Cycle structures of breakpoint graphs have rich combinatorial properties and connection to other combinatorial objects such as Bell polynomials.
Recently it was discovered that some special breakpoint graphs also appear in quantum informatics, while evaluating sums arising in the Weingarten calculus in the asymptotic limit of the number of degrees of freedom. This enabled us to employ techniques and results originally developed in bioinformatics to gain fine-grained understanding of the entanglement properties of a set of quantum states, which pose great interest to the quantum information community. Along this way we have derived PDEs for the cycle-structure infinitely-variate generating functions of the special breakpoint graphs, and further obtained and solved bivariate PDEs involving Catalan numbers, resulting in evaluation of the two leading terms in the Taylor expansion of the average entanglement entropy for random Gaussian bosonic states.
In this talk I will present the mathematical side of the story, with just a bit of bio- and quantum informatics. This is a joint work with Adam Ehrenberg, Joseph Iosue, and Alexey Gorshkov from the University of Maryland.