This study explores the effects of introducing a symmetry-breaking disorder on the dynamics of a system invariant under particle permutation. The disorder forces quantum states, confined to the $N+1$-dimensional completely symmetric space to penetrate the exponentially large $2^N$-dimensional Hilbert space of $N$ particles. In particular, we focus on the quantum kicked top as a Floquet system of $N$ qubits and use linear entropy, measuring single-qubit entanglement, to investigate the changes in the timescales and values of saturation when disorder is introduced. In the near-integrable regime of the kicked top, we study the robustness of quantum revivals to disorder. We also find that a classical calculation yields the quantum single-qubit entanglement to remarkable accuracy in the disorder-free limit. The disorder, on the other hand, is modeled in the form of noise, which again fits well with the numerical calculations. We measure the extent to which the dynamics is retained within the symmetric subspace and its spreading to the full Hilbert space using different quantities. We show that increasing disorder drives the system to a chaotic phase in full Hilbert space, as also supported by the spectral statistics. We find that there is robustness to disorder in the system, and this is a function of how chaotic the kicked top is.

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• Quantum chaos
• Quantum information theory