TITLE: Is there a quantum analog of Kolmogorov Complexity?
QUESTION [5 upvotes]: Kolmogorov Complexity (interpreted in terms of shortest program computing a string) and Shannon Entropy are quite similar.
Since there is a quantum entropy is it reasonable to ask if there is quantum analog of Kolmogorov complexity that gives rise to some sort of shortest 'quantum' program interpretation or another suitable interpretation that corresponds to quantum entropy?
What is the right analogy and the right equivalence to quantum entropy?

REPLY [2 votes]: Markus Mueller's Phd thesis is about quantum Kolmogorov complexity.
Quantum Kolmogorov Complexity and the Quantum Turing Machine
Here is his definition:
Given a Quantum Turing Machine $M$ and a finite error $\delta>0$, the finite-error quantum Kolmogorov complexity of a qubit string $|x\rangle$ is
$K_\delta^Q(x)=\min\limits_p\Big\{\ell(p): \|x-M(p)\|_{\mathrm{tr}}<\delta\Big\}$
and the approximate-scheme quantum Kolmogorov complexity of $|x\rangle$ is
$K^Q(x)=\min\limits_p\left\{\ell(p): \forall k\in\mathbb{N}: \|x-M(p,k)\|_{\mathrm{tr}}<\frac{1}{k}\right\}$
where $\|\cdot\|_{\mathrm{tr}}$ is the trace norm, i.e. $\|\rho-\sigma\|_{\mathrm{tr}}:=\frac{1}{2}\operatorname{Tr}\left(\sqrt{(\rho-\sigma)^\dagger(\rho-\sigma)}\right)$.
Another version of quantum Kolmogorov complexity of $|x\rangle$ with respect to quantum Turing machine $M$ is
$K^Q(x)=\min\limits_p\left\{\ell(p)+\lceil -\log\|\langle z|x\rangle\|^2\rceil: M(p)=|z\rangle\right\}$
You can find this version in Ming Li and Paul Vitanyi's book: An Introduction to Kolmogorov Complexity and Its Applications