Sector length distributions of recursively definable graph states through analytic combinatorics

TL;DR

This paper introduces a generating function approach to analyze the sector length distributions of recursively definable graph states, providing new analytical tools for quantum entanglement and error correction studies.

Contribution

It develops closed-form expressions for the sector length distributions of a broad class of graph states using analytic combinatorics, enabling easier analysis of their quantum properties.

Findings

Derived analytical expressions for sector length distributions of various graph states.

Provided bounds on depolarizing fidelity and multipartite entanglement criteria.

Connected generating functions with quantum information measures.

Abstract

The sector length distribution or Shor-Laflamme distribution (SLD) of quantum states is governed by the $k$-body correlations amongst the different systems, and has been used to study entanglement and error correction. A succinct description of a quantum state's SLD can be obtained by representing it through the coefficients of an appropriate weight enumerator polynomial, yielding bounds on fidelity under depolarizing noise and on multipartite entanglement. However, such expressions quickly grow out of hand and are generally difficult to achieve analytically, reflecting the computational hardness of the SLD. We sidestep this problem and, instead of a single state's SLDs, encode a family of quantum state's SLD as a generating function. We then find closed-form expressions for a large class of graph states which we call `recursively definable' and which include many common graphs such as path graphs, cycle graphs, star graphs, grid graphs, and more. As direct corollary, we obtain analytical expressions for such graph states' concentratable entanglement, bounds on their depolarizing fidelity, and a multipartite entanglement criterion. Our work opens up the use of generating functions and more generally analytic combinatorics to solve problems in quantum information theory.