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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.

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

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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.

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