# A form of entropy for pure states?

A previous question that I asked made me realize I had some major misconceptions about entropy. If I now understand this correctly, given the density matrix of a system, we can calculate Von Neumann Entropy, Renyi Entropy, or whatever other type of entropy we would like. These measures of entropy are all some sort of measure of how uncertain or disordered a system is. We expect that, for a pure state, the entropy should be 0. (This can be seen mathematically since we often return something along the lines of $$1 \times log(1) = 0$$.

However, this definition of entropy works for ensembles, not for single states.

I'm wondering if there is some notion of entropy (or any sort of disorderedness) wavefunctions. For example, if I have two wavefunctions:

$$|\psi \rangle = \sum_{n=0}^{N-1} \frac{1}{N}|n \rangle$$ $$| \phi \rangle = |l \rangle$$

(where $$l$$ is some non-negative integer), I might glance at $$|\psi \rangle$$ and say that it is "more disordered" than $$| \phi\rangle$$. However, if we were to calculate Von Neumann entropy for these pure states, we would get:

$$\rho_{|\psi\rangle} = |\psi \rangle \langle \psi |$$ $$\rho_{|\phi\rangle} = |\phi \rangle \langle \phi |$$ So therefore, for both of these cases:

$$S_{VN} = - Tr(\rho \space log (\rho)) = -Tr( \rho \space log(1) |\psi \rangle \langle \psi|) = -Tr(0) = 0$$

So Von Neumann entropy (and also Renyi entropy) don't seem to be a good measure for this.

Is there some sort of correlation function that would tell me how disordered or "spread out" a pure state is?

• Entropy isn't really meant to be a measure of disorder. If I'm reading the question correctly, you're asking if there are ways to quantify the complexity of a pure state, whether or not it corresponds to anything that would normally be called entropy. (The answer is yes, given some additional conditions on the model.) Is that a correct interpretation of the question? Aug 8 '20 at 1:07
• @ChiralAnomaly yes I think that's a much better way to ask the question! Aug 8 '20 at 1:11
• You might be interested in the first lecture in Susskind's Three Lectures on Complexity and Black Holes. Aug 8 '20 at 1:59
• If you have a preferred basis, just look at the entanglement entropy of the reduced density matrix wrt that basis. Aug 8 '20 at 2:55

If $$|\psi\rangle$$ and $$|\phi\rangle$$ had different entropies, then the entropy would depend on the basis, that is, if I transform $$|\psi\rangle$$ to a new basis:
$$|k\rangle \equiv \frac 1 A \sum_{n=0}^{N-1}\frac{2^{nk}} N|n\rangle$$
$$|\psi\rangle = |0\rangle$$