Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $O$ be an observable on a Hilbert space $\mathcal{H}$, and let $B$ be a subset of the spins composing $\mathcal{H}$, and let $\bar{B}$ be its complement. Now define

$\displaystyle O_B = \frac{1}{\operatorname{Tr}_{\bar{B}}\mathbf{1}_{\bar{B}}} \operatorname{Tr}_{\bar{B}}(O) \otimes \mathbf{1}_{\bar{B}}$.

Is this quantity equal to

$\displaystyle \int d\mu(U) U O U^\dagger$?

The integral is taken over the set of unitary operators acting on $\bar{B}$ and $\mu$ is the Haar measure of $U$. If so, why is this the case?

What physics course/book/reference introduces these sorts of integrals?

Note: this question came up from trying to understand the following paper:

share|cite|improve this question

It looks plausible to me, for two reasons:

1) The resulting expression must be invariant under conjugation by any unitary on $B$, because integration was by the Haar measure. So for product states the result of the integral has to be of the form $\rho_A \otimes \mathbf{1}_B$.

2) The integral is a linear superoperator. So what I said about product states can be extended to non-product states.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.