  We introduce the notions of algorithmic mutual information and rarity of
quantum states. These definitions enjoy conservation inequalities over unitary
transformations and partial traces. We show that a large majority of pure
states have minute self algorithmic information. We provide an algorithmic
variant to the no-cloning theorem, by showing that only a small minority of
quantum pure states can clone a non negligible amount of algorithmic
information. We also provide a chain rule inequality for quantum algorithmic
entropy. We show that rarity does not increase under POVM measurements.
