How many bits specify a quantum state?

Quantum mechanics suggests that nature is discrete, with one state per phase space volume $\hbar^{3N}$. This appears to contradict the idea that the state of an N-particle system can have infinite precision and is described by a set of exponentially many complex numbers. Using a finite-temperature gas confined in a box as an example, this short paper argues that there are indeed limits to the precision of wave functions, and that this may help at understanding the quantum-to-classical transition.