Similarity of probability amplitude functions

Question

Let's say I have two probability amplitude functions given by $\psi_1$ and $\psi_2$. That is, $\psi_i:\Sigma\rightarrow\mathbb{C}$ for some domain $\Sigma$ with $\int_\Sigma|\psi_i|^2=1$ for $i\in\{1,2\}$. Is there a canonical distance metric that can measure how "similar" these functions are?

I'm thinking of something similar to the Wasserstein or "Earth Mover's" metric for probability distributions. The $L^2$ distance (subtract, square, and integrate) doesn't work here since $\psi$ and $c\psi$ are the "same" in a quantum-mechanical sense for all $c\in\mathbb{C}$ with $|c|=1$.

[This is a follow-on to my other question]

Answer 1

The canonical metric on $CP^n$ is the Fubini-Study metric.

The distance between two states $\left| x \right\rangle$ and $\left| y \right\rangle$ is $$\gamma(x,y) = \arccos \sqrt{\frac{\left| \left\langle x \middle | y \right\rangle \right|^2}{\left\langle x \middle | x \right\rangle \left\langle y \middle | y \right\rangle}}. $$

The infinitesmal metric is thus: $$ds = \frac{\langle dx | dx \rangle}{\langle x | x \rangle} - \frac{\left | \langle dx | x \rangle \right|^2}{\left | \langle x | x \rangle \right|^2}.$$

Notice that for $CP^1$ this reduces to the natural metric on the Bloch sphere.

Answer 2

Yes, of course, the inner product itself is a canonical measure of the similarity of two wave functions – or state vectors in general. Assuming that the two wave functions are normalized to one as you wrote, the inner product $$\int_\Sigma \psi^*_1 \psi_2 $$ is a complex number whose absolute value is between $0$ and $1$. The closer the absolute value of the inner product is to one, the more "similar" the wave functions are. Let me mention that this is the genuine "physical" similarity that is unaffected by multiplying either of the wave functions by an overall phase: such overall redefinitions have no physical impact on the properties of the state.

If you wanted a compact formula for a real quantity that measures the (squared) "distance" and that vanishes if the two wave functions are the same up to the phase, you may use $$ 1 - \left|\int_\Sigma \psi^*_1 \psi_2\right|^2 $$ For a more general normalization, the first term $1$ is really $$\int_\Sigma |\psi_1|^2 \cdot \int_\Sigma |\psi_2|^2$$