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.

翻译：我们考虑的是,来自由多个异常基础组成的联盟的多余词典信号的表达很少。由多诺霍和埃拉德带来的火花在很少的表述中起着重要作用。然而,对火花的数字计算一般都是组合式的。对于几个异常基础的结合,在以前的工作中通过相互一致确定了两个较低的火花界限。我们建设性地证明这两个词都很紧。我们的主要结果为Gribonval和Nielsen在混杂基础的联盟中鲜少出现的问题提供了积极的答案。建设性证据依赖于一个相互公正的基础家庭,而这个家庭首先出现在量子信息理论中。