  We analyze the qualitative properties and the order of convergence of a
splitting scheme for a class of nonlinear stochastic Schr\"odinger equations
driven by additive It\^o noise. The class of nonlinearities of interest
includes nonlocal interaction cubic nonlinearities. We show that the numerical
solution is symplectic and preserves the expected mass for all times. On top of
that, for the convergence analysis, some exponential moment bounds for the
exact and numerical solutions are proved. This enables us to provide strong
orders of convergence as well as orders of convergence in probability and
almost surely. Finally, extensive numerical experiments illustrate the
performance of the proposed numerical scheme.
