Entanglement in selected Binary Tree States: Dicke/Total spin states, particle number projected BCS states

TL;DR

This paper investigates the entanglement properties of Binary Tree States, including Dicke and projected BCS states, providing methods to compute their entanglement entropy and analyzing their scaling and correlation characteristics.

Contribution

It introduces a practical method to compute the von Neumann entanglement entropy for subsets of qubits in Binary Tree States, with applications to Dicke and projected BCS states.

Findings

Entanglement entropy scales with subsystem size.

Upper bounds for entanglement entropy are established.

Entropies correlate with quantum fluctuations.

Abstract

Binary Tree States (BTS) are states whose decomposition on a quantum register basis formed by a set of qubits can be made sequentially. Such states sometimes appear naturally in many-body systems treated in Fock space when a global symmetry is imposed, like the total spin or particle number symmetries. Examples are the Dicke states, the eigenstates of the total spin for a set of particles having individual spin $1/2$, or states obtained by projecting a BCS states onto particle number, also called projected BCS in small superfluid systems. Starting from a BTS state described on the set of $n$ qubits or orbitals, the entanglement entropy of any subset of $ k$ qubits is analyzed. Specifically, a practical method is developed to access the $k$ qubits/particles von Neumann entanglement entropy of the subsystem of interest. Properties of these entropies are discussed, including scaling properties, upper bounds, or how these entropies correlate with fluctuations. Illustrations are given for the Dicke state and the projected BCS states.