  Akemann and Weaver showed Lyapunov-type theorem for rank one positive
semidefinite matrices which is an extension of Weaver's KS$_2$ conjecture that
was proven by Marcus, Spielman, and Srivastava in their breakthrough solution
of the Kadison-Singer problem. They conjectured that a similar result holds for
higher rank matrices. We prove the conjecture of Akemann and Weaver by
establishing Lyapunov-type theorem for trace class operators. In the process we
prove a matrix discrepancy result for sums of hermitian matrices. This extends
rank one result of Kyng, Luh, and Song who established an improved bound in
Lyapunov-type theorem of Akemann and Weaver.
