# Probability distribution of the overlap between two random quantum states in an $n$-dimensional Hilbert space? [closed]

Let two pure states $$|\Psi\rangle$$ and $$|\Phi\rangle$$ be drawn uniformly and independently from an n-dimensional Hilbert space $$\mathbb{H}^n$$. (see Note).

What is the probability distribution of the overlap $$|\langle\Psi|\Phi\rangle|^2$$?

I intuit that the mean should be $$1/n$$ but I don't know how to start working on the question of the general distribution.

Note: If you have the book "Geometry of Quantum States" by Ingemar Bengtsson and Karol Zyczkowski, you will see how such a uniform distribution can be generated. A few ways are:

• 1) Taking the first row (column) of a random unitary matrix $$U$$ distributed according to the Haar measure on $$U(N)$$.
• 2) Taking an eigenvector of a random Hermitian (unitary) matrix pertaining to GUE or CUE and multiplying it with a random phase.

etc. But unfortunately, I am not a mathematician and do not wish to go through a lot of hard work studying how to answer my question.

• I too have doubts on the fact that this makes sense, moreover, what about the phases? Are them randomly selected too? Commented Jul 17, 2019 at 14:47

Your state vectors are drawn uniformly and independently from the unit sphere of the Hilbert space (a sphere of dimension $$2n-1$$), since they are normalized states. By symmetry, you can fix one of them to be some arbitrary unit vector in the space, say $$(1,0,...,0)$$. So the question reduces to a question about the distribution of the square of the first coordinate of a uniformly random vector in the $$2n-1$$ dimensional sphere. Without the square, the answer to that can be found here (replacing $$n$$ in that post with $$2n-1$$). Figuring out the effect of the square on the probability density function is a trivial exercise.