Quantum Rotatability

TL;DR

This paper extends free probability theory by characterizing infinite quantum orthogonally and unitarily invariant sequences of noncommutative random variables as operator-valued free semicircular and circular families, respectively.

Contribution

It establishes a free analogue of Freedman's theorem for quantum invariance, linking invariance under quantum transformations to specific free probabilistic structures.

Findings

Infinite quantum orthogonal invariance implies operator-valued free semicircularity.

Quantum unitary invariance implies operator-valued free circularity.

Finite sequences do not necessarily exhibit these properties, but approximations are provided.

Abstract

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.