  Reduction of a state of a quantum system to a subsystem gives partial quantum
information about the true state of the total system. Two subalgebras A1 and A2
of B(H) are called complementary if the traceless subspaces of A1 and A2 are
orthogonal (with respect to the Hilbert-Schmidt inner product). When both
subalgebras are maximal Abelian, then the concept reduces to complementary
observables or mutually unbiased bases. In the paper several characterizations
of complementary subalgebras are given in the general case and several examples
are presented. For a 4-level quantum system, the structure of complementary
subalgebras can be described very well, the Cartan decomposition of unitaries
plays a role. It turns out that a measurement corresponding to the Bell basis
is complementary to any local measurement of the two-qubit-system.
