We establish a connection between the problem of constructing maximal collections of mutually unbiased bases (MUBs) and an open problem in the theory of Lie algebras. More precisely, we show that a collection of
μ MUBs in $\K^n$ gives rise to a collection of
μ Cartan subalgebras of the special linear Lie algebra $sl_n(\K)$ that are pairwise orthogonal with respect to the Killing form, where $\K=\R$ or $\K=\C$. In particular, a complete collection of MUBs in $\C^n$ gives rise to a so-called orthogonal decomposition (OD) of $sl_n(\C)$. The converse holds if the Cartan subalgebras in the OD are also
†-closed, i.e., closed under the adjoint operation. In this case, the Cartan subalgebras have unitary bases, and the above correspondence becomes equivalent to a result of \cite{bbrv02} relating collections of MUBs to collections of maximal commuting classes of unitary error bases, i.e., orthogonal unitary matrices. This connection implies that for
n≤5 an essentially unique complete collection of MUBs exists. We define \emph{monomial MUBs}, a class of which all known MUB constructions are members, and use the above connection to show that for
n=6 there are at most three monomial MUBs.
Let q be a power of 2. We show by representation theory that there exists a q x q unitary matrix of multiplicative order q+1 whose powers generate q+1 pairwise mutually unbiased base in C^q. When q is a power of an odd prime, there is a q x q unitary matrix of multiplicative order q+1 whose first (q+1)/2 powers generate (q+1)/2 pairwise mutually unbiased bases. We also show how the existence of these matrices implies the existence of a special type of orthogonal decomposition with respect to the Killing form of the special linear and symplectic Lie algebras.
There has been great interest in finding sets of m mutually unbiased bases which are compatible with a given space C, specially in physics due to their interesting applications in quantum information theory. Several general results have been obtained so far, but surprising results may occur for definite (m, d)-values. One such case that has remained an open question (the simplest case) is the one regarding the existence of m = 4 mutually orthogonal bases for d = 6. In the present work we introduce a new approach to the problem by translating it into an optimization procedure for a given pair (m, d).
There has been great interest in finding sets of
m mutually unbiased bases which are compatible with a given space
Cd, specially in physics due to their interesting applications in quantum information theory. Several general results have been obtained so far, but surprising results may occur for definite
(m,d)-values. One such case that has remained an open question (the simplest case) is the one regarding the existence of
m=4 mutually orthogonal bases for
d=6. In the present work we introduce a new approach to the problem by translating it into an optimization procedure for a given pair
(m,d).
In a quantum system having a finite number N of orthogonal states, two orthonormal bases {ai} and {bj} are called mutually unbiased if all inner products ⟨ai∣bj⟩ have the same modulus 1∕N. This concept appears in several quantum information problems. The number of pairwise mutually unbiased bases is at most N+1 and various constructions of such N+1 bases have been found when N is a power of a prime number. We study families of formulas that generalize these constructions to arbitrary dimensions using finite rings. We then prove that there exists a set of N+1 mutually unbiased bases described by such formulas, if and only if N is a power of a prime number.