Prefix-free quantum Kolmogorov complexity

We introduce quantum-K ($QK$), a measure of the descriptive complexity density matrices using classical prefix-free Turing machines and show that initial segments weak Solovay random quantum Schnorr states are incompressible in sense $QK$. Many properties enjoyed by Kolmogorov ($K$) have analogous versions for $QK$; notably counting condition. Several connections between randomness $K$, including Chaitin type characterization randomness, carry over to those work towards Levin-Schnorr terms has $K_C$; version $K$ computable machine, $C$. similarly define $QK_C$, Quantum is shown $QK_C$. The latter implies $K_C$.