I was thinking about how quantum states are represented for various types of systems, and how the amount of classical information (bits) required to represent a state depends on its basis.
Let's take hydrogen for example. You can represent it in the position basis using dirac delta functions. This is probably the least efficient way to store it (for numerical work). Or you can use the exact analytical functions themselves: hydrogen is a product of the spherical harmonics and a decaying radial exponential function. Storing these symbolic equations is probably the most efficient method.
For small molecules, DFT typically represents the wavefunction as a combination of Gaussian or Slater type orbitals, and it requires a certain number of these orbitals to achieve a desired numerical accuracy (I believe fewer Slater type orbitals are strictly necessary, but Gaussian type orbitals are used in practice since they're more computationally efficient).
Heck, you could even store an arbitrary wavefunction as a list of coefficients in its Taylor expansion.
My point is: there is some relationship between the number of bits needed to represent a state vector and the accuracy of calculations performed on it (energy eigenvalues for instance), and this number seems to vary a lot.
So naturally, I wondered, what is the optimal representation? How can I store this state vector with the least number of bits possible? Is there an algorithm to determine this?
Strangely enough, I can't find any literature on the subject! It seems like this would be a very important question for practical numerical work, but nothing comes up (it seems like everyone is content just using plane waves and Gaussians for everything). Maybe I'm just using the wrong search terms. Does anyone know if there has been any work on this? Is there a theoretical absolute minimum number of bits required to represent a particular state?