Entropy Cones and Entanglement Evolution for Dicke States

Abstract

The $N$-qubit Dicke states $|D_k^N⟩$, 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 $|D_k^N⟩$ entropy cone. We demonstrate that all $|D_k^N⟩$ entropy vectors emerge symmetrized, and use this to define a min-cut protocol on star graphs which realizes $|D_k^N⟩$ entropy vectors. We identify the stabilizer group for all $|D_k^N⟩$, under the action of the $N$-qubit Pauli group and two-qubit Clifford group, which we use to construct $|D_k^N⟩$ reachability graphs. We use these reachability graphs to analyze and bound the evolution of $|D_k^N⟩$ entropy vectors in Clifford circuits.