# Distribution of partial trace of Haar unitary

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?
• 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($2^n$)? Dec 22, 2022 at 20:14
• @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) Jan 5 at 3:17
• 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? Jan 6 at 20:32