  We consider sparse representations of signals from redundant dictionaries
which are unions of several orthonormal bases. The spark introduced by Donoho
and Elad plays an important role in sparse representations. However, numerical
computations of sparks are generally combinatorial. For unions of several
orthonormal bases, two lower bounds on the spark via the mutual coherence were
established in previous work. We constructively prove that both of them are
tight. Our main results give positive answers to Gribonval and Nielsen's open
problem on sparse representations in unions of orthonormal bases. Constructive
proofs rely on a family of mutually unbiased bases which first appears in
quantum information theory.
