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Many-body states that are invariant under particle relabelling, thepermutation symmetric states, occur naturally when the system dynamics isdescribed by symmetric processes or collective spin operators. We deriveexpressions for the reduced density matrix for arbitrary subsystemdecomposition for these states and study properties of permutation symmetricstates and their subsystems when the joint system is picked randomly anduniformly. Thus defining a new random matrix ensemble, we find the averagelinear entropy and von Neumann entropy which implies that random permutationsymmetric states are marginally entangled and as a consequence the tripartitemutual information (TMI) is typically positive, preventing information frombeing shared globally. Applying these results to the quantum kicked top viewedas a multi-qubit system we find that entanglement, mutual information and TMIall increase for large subsystems across the Ehrenfest or log-time and saturateat the random state values if there is global chaos. During this time theout-of-time order correlators (OTOC) evolve exponentially implying scramblingin phase space. We discuss how positive TMI may coexist with such scrambling.

Tripartite mutual information, entanglement, and scrambling in permutation symmetric systems with an application to quantum chaos