# Similarity of probability amplitude functions

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]

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Such a "canonical distance metric" should follow from practical needs; otherwise it is vague and ambiguous. –  Vladimir Kalitvianski Oct 6 '12 at 10:57
Both of the answers below provide useful metrics. Is there a metric that -- like the Wasserstein metric -- uses underlying distances on the space? –  Justin Solomon Oct 7 '12 at 15:28

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.

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Ah, I think this is exactly what I need! BTW, this document appeals nicely to us math folks: physik.uni-leipzig.de/~uhlmann/PDF/UC07.pdf –  Justin Solomon Oct 9 '12 at 21:46
Incidentally, the pullback of this metric on families of ground states of Hamiltonians gives a nice way to geometrically describe critical behaviour. E.g. Venuti and Zanardi, 2007. –  genneth Oct 10 '12 at 3:22
Interesting -- I'll take a look, this seems like the type of physics I might be able to follow :-) –  Justin Solomon Oct 11 '12 at 0:53

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 $$\int_\Sigma |\psi_1|^2 \cdot \int_\Sigma |\psi_2|^2$$

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This doesn't quite respect the "natural" symmetries of a $CP^n$ space... –  genneth Oct 6 '12 at 11:50