Let $A$ be a matrix of the Gaussian unitary ensemble (GUE) and $v_1,v_2$ be two orthonormal vectors. 

I wonder if one can compute the expectation value 

$$\mathbf E\sqrt{4\langle Av_1,v_2 \rangle^2 + (\langle Av_1,v_1 \rangle-\langle Av_2,v_2 \rangle)^2}.$$

We can choose a unitary $U$ such that $Ue_1=v_1$ and $U e_2 = v_2$ then we have 

$$\mathbf E\sqrt{4\langle U^*AUe_1,e_2 \rangle^2 + (\langle U^*AUe_1,e_1 \rangle-\langle U^*AUe_2,e_2 \rangle)^2}.$$

Since the GUE is invariant under unitaries, this implies we have to compute

$$\mathbf E\sqrt{4a_{12}^2 + (a_{11}-a_{22})^2},$$

where $a_{ij}$ are the matrix entries of $A.$