  For a quantum system in a macroscopically large volume $V$, prepared in a
pure state and subject to maximally noisy or ergodic unitary dynamics, the
reduced density matrix of any sub-system $v\ll V$ is almost surely totally
mixed. We show that the fluctuations around this limiting value, evaluated
according to the invariant measure of these unitary flows, are captured by the
Gaussian unitary ensemble (GUE) of random matrix theory. An extension of this
statement, applicable when the unitary transformations conserve the energy but
are maximally noisy or ergodic on any energy shell, allows to decipher the
fluctuations around canonical typicality. According to typicality, if the large
system is prepared in a generic pure state in a given energy shell, the reduced
density matrix of the sub-system is almost surely the canonical Gibbs state of
that sub-system. We show that the fluctuations around the Gibbs state are
encoded in a deformation of the GUE whose covariance is specified by the Gibbs
state. Contact with the eigenstate thermalisation hypothesis (ETH) is
discussed.
