Entanglement in selected binary tree states: Dicke or total spin states or particle-number-projected BCS states

Binary tree states (BTSs) 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, such as 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 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 $k$-qubit or $k$-particle 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.