  In arXiv:0807.0677, K\"ostler and Speicher observed that de Finetti's theorem
on exchangeable sequences has a free analogue if one replaces exchangeability
by the stronger condition of invariance under quantum permutations. In this
paper we study sequences of noncommutative random variables whose joint
distribution is invariant under quantum orthogonal transformations. We prove a
free analogue of Freedman's characterization of conditionally independent
Gaussian families, namely an infinite sequence of self-adjoint random variables
is quantum orthogonally invariant if and only if they form an operator-valued
free centered semicircular family with common variance. Similarly, we show that
an infinite sequence of noncommutative random variables is quantum unitarily
invariant if and only if they form an operator-valued free centered circular
family with common variance.
  We provide an example to show that, as in the classical case, these results
fail for finite sequences. We then give an approximation to how far the
distribution of a finite quantum orthogonally invariant sequence is from that
of an operator-valued free centered semicircular family with common variance.
