Quasi-orthogonal Subalgebras of Matrix Algebras@

In the theory of quantum mechanics, an n-level system is described by the algebra Mn =Mn(C) of n× n complex matrices. The matrix algebra of a composite system consisting of an n-level and an m-level system is Mn ⊗Mm ≃ Mnm. A subalgebra of Mk corresponds to a subsystem of a k-level quantum system. In this paper, subalgebras contain the identity and are closed under the adjoint operation of matrices, that is, they are unital ∗-subalgebras. The algebra Mn can be endowed by the inner product 〈A,B〉 = Tr(AB) and it becomes a Hilbert space. Two subalgebras A1 and A2 are called quasi-orthogonal if A1 ⊖ CI ⊥ A2 ⊖ CI. We consider pairwise quasi-orthogonal subalgebras A1,A2, . . . ,Al in Mpkn which are isomorphic to Mpk for k ≥ 1, n ≥ 2 and a prime number p with p ≥ 3. The aim of this paper is to obtain the maximum l. The case p = 2, n = 2 and k = 1 is shown in [3, 6, 8] and the maximum is 4. The motivations of this problem are followings. If a total system Mn ⊗ Mm has a statistical operator ρ, we can reconstruct the reduced density ρ (1) n = Trmρ in the subsystem Mn, where Trm is a partial trace onto Mn. In order to get more information, we change the density ρ by an interaction. For a Hamiltonian H , the new state is eρe =W1ρW ∗ 1