  In this work, Bernstein's concentration inequalities for squared integrable
matrix-valued discrete-time martingales are obtained. Based on Lieb's theory
and Bernstein's condition, a suitable supermartingale can be constructed. Our
proof is largely based on this new exponential supermartingale, Freedman's
method, and Doob's stopping theorem. Our result can be regarded as an extension
of Tropp's work (ECP, 2012).
