  We demonstrate that the concept of quantum typicality allows for significant
progress in the study of real-time spin dynamics and transport in quantum
magnets. To this end, we present a numerical analysis of the spin-current
autocorrelation function of the antiferromagnetic and anisotropic spin-1/2
Heisenberg chain as inferred from propagating only a single pure state,
randomly chosen as a "typical" representative of the statistical ensemble.
Comparing with existing time-dependent density-matrix renormalization group
(tDMRG) data, we show that typicality is fulfilled extremely well, consistent
with an error of our approach, which is perfectly under control and vanishes in
the thermodynamic limit. In the long-time limit, our results provide for a new
benchmark for the enigmatic spin Drude weight, which we obtain from chains as
long as L=33 sites, i.e., from Hilbert spaces of dimensions almost O(10^4)
larger than in existing exact-diagonalization studies.
