The law of large numbers for discrete generalized quantum channels

We consider random linear operators $\mathrm{\Omega }\to \mathcal{L}({\mathcal{T}}_p,{\mathcal{T}}_p)$ acting in a $p$-th Schatten class ${\mathcal{T}}_p$ in a separable Hilbert space $\mathcal{H}$ for some $1⩽p<\mathrm{\infty }$. Such a superoperator is called a pre-channel since it is an extension of a quantum channel to a wider class of operators without requirements of trace-preserving and positivity. Instead of the sum of i.i.d. variables there may be considered the composition of random semigroups $e^{A_{i}t\mathrm{/}n}$ in the Banach space ${\mathcal{T}}_p$. The law of large numbers is known in the case $p=2$ in the form of the usual law of large numbers for random operators in a Hilbert space. We obtain the law of large numbers for the case $1⩽p⩽2$.