  We study the compression of n quantum systems, each prepared in the same
state belonging to a given parametric family of quantum states. For a family of
states with f independent parameters, we devise an asymptotically faithful
protocol that requires a hybrid memory of size (f/2)log(n), including both
quantum and classical bits. Our construction uses a quantum version of local
asymptotic normality and, as an intermediate step, solves the problem of
compressing displaced thermal states of n identically prepared modes. In both
cases, we show that (f/2)log(n) is the minimum amount of memory needed to
achieve asymptotic faithfulness. In addition, we analyze how much of the memory
needs to be quantum. We find that the ratio between quantum and classical bits
can be made arbitrarily small, but cannot reach zero: unless all the quantum
states in the family commute, no protocol using only classical bits can be
faithful, even if it uses an arbitrarily large number of classical bits.
