2
$\begingroup$

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?
$\endgroup$
3
  • $\begingroup$ 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($2^n$)? $\endgroup$ 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, 2023 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? $\endgroup$ Jan 6, 2023 at 20:32

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.