  Entanglement entropy of typical quantum states, also known as the Page curve,
plays an important role in quantum many-body systems and quantum gravity.
However, little has hitherto been understood about the role of symmetry in
quantum entanglement. Here, we establish the comprehensive classification of
typical quantum entanglement for free fermions, or equivalently the quadratic
Sachdev-Ye-Kitaev model with symmetry, on the basis of the tenfold fundamental
symmetry classes of time reversal, charge conjugation, and chiral
transformation. Through both analytical and numerical calculations of random
matrix theory, we show that the volume-law contribution to average entanglement
entropy is robust and remains unaffected by symmetry. Conversely, we uncover
that the constant terms of the average and variance of entanglement entropy
yield tenfold universal values unique to each symmetry class. These constant
terms originate from the combination of a global scaling of the entanglement
spectrum due to time-reversal symmetry and a singular peak at the center of the
entanglement spectrum due to chiral or particle-hole symmetry. Our work
elucidates the interplay of symmetry and entanglement in quantum physics and
provides characterization of symmetry-enriched quantum chaos.
