Currently, I am going through Algorithmic complexity of quantum states by Mora et al.In Section $2$, on page $5$, the authors discusses the finding of a natural upper bound also for the complexity of a quantum state. To do that, the authors introduce the concept of volume of a particular set. They consider the set of all infinite binary strings which is isomorphic to the unit interval $[0, 1]$. So, the volume is defined as follows.Any $n$-bit binary string $\omega_n$ identifies the set of infinite strings whose first $n$ bits coincide with $\omega_n$: the volume of this set (a ball, $B_{\omega_n}$) is $V(B_{\omega_n}) \sim 2^{-n}$.Could anyone please explain to me why it is $\sim 2^{-n}$?
natural upper bound also for the complexity of a quantum state
cc.complexity theory;quantum computing;it.information theory;quantum information;kolmogorov complexity