Martin-Löf random quantum states

We extend the key notion of Martin-Löf randomness for infinite bit sequences to the quantum setting, where the sequences become states of an infinite dimensional system. We prove that our definition naturally extends the classical case. In analogy with the Levin-Schnorr theorem, we work toward characterizing quantum ML-randomness of states by incompressibility (in the sense of quantum Turing machines) of all initial segments.