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I am sure this must have been covered in the mathematical literature, 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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  • $\begingroup$ Do you know what the answer is for $k=1$? $n=2$? Also by "the Haar measure" I presume you mean the circular unitary distribution for SU($2^n$)? $\endgroup$
    – Quantum Mechanic
    Dec 22, 2022 at 20:14
  • $\begingroup$ @QuantumMechanic I can calculate low moments of the random variable, but it would be nice to get the full distribution. And yes, unitaries drawn from the CUE for matrices of dim $2^n$. The standard name this uniform distribution is called the Haar measure though (this is standard terminology in quantum computing and math) $\endgroup$
    – nervxxx
    Jan 5 at 3:17
  • $\begingroup$ Can you say anything about the distribution of the trace or the determinant? The CUE tells us the distribution of eigenvalues of $U$, and the eigenvalues of $W$ must sum to the eigenvalues of $U$, so we can write the distribution of the trace of $W$. The determinants won't be the same, but maybe there's some constraint? $\endgroup$
    – Quantum Mechanic
    Jan 6 at 20:32

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