Trivial Mathematics
There is a small(?) difference that this reasoning makes apparent: in the quantum setting, if you are given an infinite stream of i.i.d.-and-unentangled impure states, you can't estimate the probability distribution they come from. That's because states with the same density matrix provide same distributions on outcomes of any measurements (and the density matrix of the i.i.d. sequence is a function of the density matrix of a single element).
Trivial Mathematics
By impure state I meant a probability distribution over (pure) states. Consider two probability distributions over single-qubit states: one of them uniform (i.e. invariant under any unitary transformation) and the other one assigning probability 1/2 to |0> and to |1>. Any measurement made on these will provide same probability distribution of outcomes, because they have the same density matrix (density matrix is expected value of the projection operator onto the (pure) state, where the expectation ranges over the probability distribution from which the state is drawn).
Trivial Mathematics
@robryk Oh yes, impure states can be indistinguishable from each other. But experiments can recover the elements of the density matrix even if not the probability distribution from which pure states were selected.
Trivial Mathematics
@dpiponi Precisely; for each entry of that matrix you can conduct an experiment, outcome of which has an expected value equal to that entry.
I think that this is an important observation when trying to map classical and quantum concepts: probability distributions do not map into pure states or into probability distributions over states, but into density matrices. (I emphasize this, because when I first encountered the concept of density matrices that wasn't made very obvious and it was later an epiphany to me that density matrices encode precisely all the information that's available via experiments (on infinite exchangeable sequences of states)).
Trivial Mathematics
@robryk You can observe 100 qubits (say), then perform a unitary rotation and observe another 100. Then you can estimate everything (up to phase) can't you?
I think that formally corresponds to a procedure I've used in real life where you take multiple photos with different polarizations so as to remove specular reflections (which tend to be polarized).