  Due to M\"{u}ller's theorem, the Kolmogorov complexity of a string was shown
to be equal to its quantum Kolmogorov complexity. Thus there are no benefits to
using quantum mechanics to compress classical information. The quantitative
amount of information in classical sources is invariant to the physical model
used. These consequences make this theorem arguably the most important result
in the intersection of algorithmic information theory and physics. The original
proof is quite extensive. This paper contains two simple proofs of this
theorem. This paper also contains new bounds for quantum Kolmogorov complexity
with error.
