arXiv subjects

Adam Burchardt

Publications and source records attributed to Adam Burchardt.

21 records · Page 2Linked to original sources

arXiv2019

Cut-resistant links and multipartite entanglement resistant to particle loss

In this work, we explore the space of quantum states composed of $N$ particles. To investigate the entanglement resistant to particles loss, we introduce the notion of $m$-resistant states. A quantum state is $m$-resistant if it remains entangled after losing an arbitrary subset of m particles, but becomes separable after losing a number of particles larger than $m$. We establish an analogy to the problem of designing a topological link consisting of $N$ rings such that, after cutting any $(m + 1)$ of them, the remaining rings become disconnected. We present a constructive solution to this problem, which allows us to exhibit several distinguished $N$-particles states with the desired property of entanglement resistance to a particle loss.

quant-ph↗

arXiv2018

Algebras with two multiplications and their cumulants

Cumulants are a notion that comes from the classical probability theory, they are an alternative to a notion of moments. We adapt the probabilistic concept of cumulants to the setup of a linear space equipped with two multiplication structures. We present an algebraic formula which involves those two multiplications as a sum of products of cumulants. In our approach, beside cumulants, we make use of standard combinatorial tools as forests and their colourings. We also show that the resulting statement can be understood as an analogue of Leonov--Shiraev's formula. This purely combinatorial presentation leads to some conclusions about structure constant of Jack characters.

math.CO↗

arXiv2018

The top-degree part in the Matchings-Jack Conjecture

In 1996 Goulden and Jackson introduced a family of coefficients $( c_{π, σ}^λ ) $ indexed by triples of partitions which arise in the power sum expansion of some Cauchy sum for Jack symmetric functions $( J^{(α)}_π)$. The coefficients $ c_{π, σ}^λ $ can be viewed as an interpolation between the structure constants of the class algebra and the double coset algebra. Goulden and Jackson suggested that the coefficients $ c_{π, σ}^λ $ are polynomials in the variable $β:= α-1$ with non-negative integer coefficients and that there is a combinatorics of matching hidden behind them. This \emph{Matchings-Jack Conjecture} remains open. Doł\oldk{e}ga and Féray showed the polynomiality of connection coefficients $c^λ_{π,σ}$ and gave the upper bound on the degrees. We give a necessary and sufficient condition for the polynomial $ c_{π, σ}^λ$ to achieve this bound. We show that the leading coefficient of $ c_{π, σ}^λ$ is a positive integer and we present it in the context of Matchings-Jack Conjecture of Goulden and Jackson.

math.CO↗