  We study the formation and evolution of Brightest Cluster Galaxies starting
from a $z=2$ population of quiescent ellipticals and following them to $z=0$.
To this end, we use a suite of nine high-resolution dark matter-only
simulations of galaxy clusters in a $\Lambda$CDM universe. We develop a scheme
in which simulation particles are weighted to generate realistic and
dynamically stable stellar density profiles at $z=2$. Our initial conditions
assign a stellar mass to every identified dark halo as expected from abundance
matching; assuming there exists a one-to-one relation between the visible
properties of galaxies and their host haloes. We set the sizes of the luminous
components according to the observed relations for $z\sim2$ massive quiescent
galaxies. We study the evolution of the mass-size relation, the fate of
satellite galaxies and the mass aggregation of the cluster central. From $z=2$,
these galaxies grow on average in size by a factor 5 to 10 of and in mass by 2
to 3. The stellar mass growth rate of the simulated BCGs in our sample is of
1.9 in the range $0.3<z<1.0$ consistent with observations, and of 1.5 in the
range $0.0<z<0.3$. Furthermore the satellite galaxies evolve to the present day
mass-size relation by $z=0$. Assuming passively evolving stellar populations,
we present surface brightness profiles for our cluster centrals which resemble
those observed for the cDs in similar mass clusters both at $z=0$ and at $z=1$.
This demonstrates that the $\Lambda$CDM cosmology does indeed predict minor and
major mergers to occur in galaxy clusters with the frequency and mass ratio
distribution required to explain the observed growth in size of passive
galaxies since $z=2$. Our experiment shows that Brightest Cluster Galaxies can
form through dissipationless mergers of quiescent massive $z=2$ galaxies,
without substantial additional star formation.
