The Central Limit Theorem for random exponents on a Hilbert space in the Weak Operator Topology

Abstract

We consider random linear continuous operators $\Omega \to \mathcal{L}(\mathcal{H}, \mathcal{H})$ on a Hilbert space $\mathcal{H}$. For example, such random operators may be random quantum channels. The Central Limit Theorem is known for the sums of i.i.d. random operators. Instead of the sum, there may be considered the composition of random exponents $e^{A_i/n}$. We obtain the Central Limit Theorem in the Weak Operator Topology for centralized and normalized random exponents of i.i.d. linear continuous operators on a Hilbert space.