Minimum information necessary to represent a pure quantum state

Question

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?

Answer 1

While I am not versed in the wave functions of particles, I can add perhaps a little bit of intuition as to the approach you are taking to this process. What you are asking for sounds a lot like a common question in number theory and computer science: what is the Kolmogorov Complexity of a given piece of data? In other words, what is the most efficient way to describe a set of data, marking efficiency as the relative smallness of the description? Unfortunately, there are a few paradoxes here unless you have a very concrete definition of "definition". A good example would be the Berry Paradox.

Answer 2

Let me first quickly answer what I believe will be a misunderstanding of your question: a pure quantum state has no Shannon entropy, in the sense it can be treated as a known point in the spectrum of an observable: you can think of this spectrum as an alphabet of symbols and knowing the pure quantum state is tantamount to knowing which symbol we have. A classical mixture of quantum states, such as in a classically random distribution of different pure quantum states or a mixture arising in the Wigner's Friend thought experiment has Shannon entropy given by the von Neumann entropy (see Wiki page of this name).

But what I think you're seeking is, as Speleo's answer states, an information content given by the shortest possible description of a quantum state in a target language. This, as Speleo states, is the Kolmogorov complexity, and it is always defined relative to a particular language. For the last part of my answer, the following result is important:

Theorem: Given any language $\mathscr{S}$, there is no finite algorithm definable in that language that can compute the Kolmogorov complexity $\mathscr{K}:\mathscr{A}\times \mathscr{A}\times\,\cdots\,\times\,\mathscr{A}\to\mathbb{N}$ of a general string $s\in \mathscr{A}\times \mathscr{A}\times\,\cdots\,\times\,\mathscr{A}$ in a specified alphabet $\mathscr{A}$

The easiest and commonest proof is given on my website here. It is the only uncomputability proof that I know of which does not call on the Cantor Slash argument.

Now the only work that I have seen which is vaguely relevant to your question is the highly interesting speculation made by Charles Bennett in

Charles Bennett, "The Thermodynamics of Computation: A Review", Int. J. Theo. Phys., 21, No. 12, 1982)

Late in this paper, he makes the interesting observation that, because the Kolmogorov complexity is uncomputable, then there is no algorithm for computing the shortest description of a chemical arrangement (which can be given in some language encoding the relative spatial relationships between a molecule's constituents, for example). Bennett and others believe that the second law of thermodynamics, at least in part, arises through the fundamental reversibility (NOTE I did NOT say IRReversibility!) of physics at the microscopic level, thus obliging a Szilard engine to encode all its history in the microstate of the environment (and thus encode the former history of the heat reservoir in the environment). Thus one must do work the "throw" the excess entropy out of the immediate hardware of the Maxwell Daemon computer within a Szilard engine, so that we "re-intialise" it ready to encode more history. I discuss this idea further in my answer here.

So now, if this idea - essentially a restatement of Landauer's principle - underlies the second law, then we need to know how Nature encodes Her states to say how much information a system can "soak up" at a given temperature. Thus the Kolmogorov complexity would seem important for the theoretical computation of entropies of substances.

Therefore Bennett conjectures that the uncomputability of the Kolmogorov Complexity is the reason why theoretical calculations of entropies of substances in physical chemistry are notoriously unreliable.

A thoroughly fascinating idea that I think definitely has merit, but it clearly needs further work on it, particularly in making it fully mathematically and probability-theoretically rigorous.