O ct 2 00 6 Complementary reductions for two qubits

Reduction of a state of a quantum system to a subsystem gives partial quantum information about the true state of the total system. In connection with optimal state determination for two qubits, the question was raised about the maximum number of pairwise complementary reductions. The main result of the paper tells that the maximum number is 4, that is, if A,A, . . . ,A are pairwise complementary (or quasi-orthogonal) subalgebras of the algebra M4(C) of all 4 × 4 matrices and they are isomorphic to M2(C), then k ≤ 4. The proof is based on a Cartan decomposition of SU(4). In the way to the main result, contributions are made to the understanding of the structure of complementary reductions.