Complementary reductions for two qubits

Abstract

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 A1, A2,... Ak are pairwise complementary (or quasi-orthogonal) subalgebras of the algebra of all 4x4$ matrices and they are isomorphic to the algebra of all 2x2 matrices, then k is at most 4. In the way to this result, contributions are made to the understanding of the structure of complementary reductions.

7 pages, key words: mutually unbiased bases, unbiased measurements, complementary subalgebras, Cartan decomposition, Pauli matrices