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I am sure this must have been covered in the mathematical literature (and is certainly related to a similar question I had asked previously), but hoping someone can direct me to the right place.

Let us be given random unitaries $U$ on $n$ qubits (so dimension of the space is $2^n$) distributed according to the Haar measure.

Define the partial trace of such unitaries on $k \leq n$ qubits, $W = \text{Tr}_{1,\cdots,k} U$, so that $W$ acts on $n-k$ qubits.

  1. Is there a nice closed form expression of the distribution of $W$ for any $n,k$? (Note in the limit when $n,k$ is large, I believe $W$ should be distributed according to the Ginibre ensemble [unless I'm mistaken], but here I'm asking for the induced distribution for arbitrary $n,k$)
  2. Is there a computationally/numerically efficient way of generating the random matrices $W$ without generating the full $U$ and performing the partial trace?
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  • $\begingroup$ are you sure it's the partial trace of the unitary matrix you seek, and not the partial trace of the density matrix of your qubits? $\endgroup$
    – Carlo Beenakker
    Commented Dec 21, 2022 at 9:26
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    $\begingroup$ for large $k$ the partial trace will be a matrix with approximately independent complex matrix elements with a normal distribution of zero mean and variance $k/n$; I would be surprised if there are exact results for small $k$. $\endgroup$
    – Carlo Beenakker
    Commented Dec 21, 2022 at 9:59
  • $\begingroup$ @CarloBeenakker Hi, thanks, yes, it's the partial trace of the unitary I'm seeking. And yes, your second comment is in line with my intuition about the Ginibre ensemble (rescaled). $\endgroup$
    – nervxxx
    Commented Dec 21, 2022 at 11:19

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