A gap for the maximum number of mutually unbiased bases (Q2838954)

Two orthonormal bases \(\mathcal{E}=\{ {\mathbf e}_1,\dots, {\mathbf e}_d \}\), \(\mathcal{F}=\{ {\mathbf f}_1,\dots, {\mathbf f}_d\}\) in \(\mathbb{C}^d\) are mutually unbiased if \(|\langle {\mathbf e}_k,{\mathbf f}_j\rangle |=1/\sqrt{d}\) for all \(k,j=1,\dots, d\). It is well known that any set of mutually unbiased orthonormal bases in \(\mathbb{C}^d\) can have at most \(d+1\) elements. The author shows that if a set of mutually unbiased orthonormal bases in \(\mathbb{C}^d\) has \(d\) elements, then there exists another orthonormal basis which is mutually unbiased with respect to the elements of that set, i.e., a \(d\)-element set of mutually unbiased bases can always be completed to a set of maximal possible cardinality. The proof proceeds via identification of the orthonormal bases in the set with \(d\) maximal abelian self-adjoint subalgebras (masas) in \(M_d(\mathbb{C})\) which, because of bases being mutually unbiased, are orthogonal (in an appropriate sense) under the Hilbert-Schmidt inner product. Then, using an interesting new criterion as to when a self-adjoint subspace in \(M_n(\mathbb{C})\) is an algebra, the common orthogonal complement of these \(d\) masas is shown to be a masa, thus stemming from another orthonormal basis. In this sense, the construction is explicit, although passing back to the orthonormal basis requires diagonalization of the masa thus obtained. Note, however, that unless \(d\) is a power of prime, the question of the existence of a set of mutually unbiased orthonormal bases of maximal possible cardinality \(d+1\), hence also \(d\), is unsettled.