Exact Quantification of Entanglement for Permutationally Invariant States in Spin Entanglement

Quantifying mixed-state entanglement of a many-body system has been a challenging task. In this work, we quantify the entanglement of states in unresolvable spin ensembles which are inherently mixed. By exploiting the permutationally invariant (PI) properties of these states, we show that the bipartite entanglement of a wide range of unresolvable ensemble states can be calculated exactly. Our formalism is versatile; it allows us to calculate the entanglement of states with an arbitrary number of particles, effective angular momentum, and bi-partition. We apply our method to explore the characteristics of entanglement in different physically motivated scenarios, including the states with a definite magnetization and metrologically useful superposition states such as the GHZ-like states and spin-squeezed states. Our method is helpful to understanding the role of entanglement in quantum technologies.