# Tracing out an observable vs integrating over unitaries

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?

Note: this question came up from trying to understand the following paper: http://arxiv.org/abs/quant-ph/0603121

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 $\rho_A \otimes \mathbf{1}_B$.