Quantum Rotatability

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.