Complementarity in quantum systems

Abstract

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.