Entropy Cones and Entanglement Evolution for Dicke States

TL;DR

This paper analyzes the entanglement entropy of Dicke states, characterizes their entropy cones, and studies their evolution under Clifford circuits using stabilizer groups and reachability graphs.

Contribution

It provides a general calculation of entanglement entropy for Dicke states, describes their entropy cone, and introduces methods to analyze their entanglement evolution.

Findings

All Dicke state entropy vectors are symmetrized.

A min-cut protocol on star graphs realizes Dicke state entropy vectors.

Stabilizer groups are identified, enabling reachability graph construction.

Abstract

The $N$-qubit Dicke states $|D^N_k\rangle$, of Hamming-weight $k$, are a class of entangled states which play an important role in quantum algorithm optimization. We present a general calculation of entanglement entropy in Dicke states, which we use to describe the $|D^N_k\rangle$ entropy cone. We demonstrate that all $|D^N_k\rangle$ entropy vectors emerge symmetrized, and use this to define a min-cut protocol on star graphs which realizes $|D^N_k\rangle$ entropy vectors. We identify the stabilizer group for all $|D^N_k\rangle$, under the action of the $N$-qubit Pauli group and two-qubit Clifford group, which we use to construct $|D^N_k\rangle$ reachability graphs. We use these reachability graphs to analyze and bound the evolution of $|D^N_k\rangle$ entropy vectors in Clifford circuits.