Searching for Mutually Unbiased Observables

Takayuki Miyadera

Berge Englert

STEPHEN D. BRIERLEY, Stefan Weigert

We study sets of pure states in a Hilbert space of dimension $d$ which are mutually unbiased (MU), that is, the moduli of their scalar products are equal to zero, one, or $1∕\sqrt{d}$. Each of these sets will be called a MU constellation, and if four MU bases were to exist for $d=6$, they would give rise to 35 different MU constellations. Using a numerical minimization procedure, we are able to identify only 18 of them in spite of extensive searches. The missing MU constellations provide the strongest numerical evidence so far that no seven MU bases exist in dimension 6.

Stanley Gudder

M. Kibler

Two equivalent ways of looking for mutually unbiased bases are discussed in this note. The passage from the search for d+1 mutually unbiased bases in Cd to the search for d(d+1) vectors in Cd2 satisfying constraint relations is clarified. Symmetric informationally complete positive-operator-valued measures are briefly discussed in a similar vein.

Franklin E. Schroeck