Golden states in resource theory of superposition

One central study that constitutes a major branch of quantum resource theory is the hierarchy of states. This provides a broad understanding of resourcefulness in certain tasks in terms of efficiency. Here we investigate the maximal superposition states, i.e., golden states, of the resource theory of superposition. Golden states in the resource theory of coherence are very well established; however, it is a very challenging task for superposition due to the nonorthogonality of the basis states. We show that there are sets of inner product settings that admit a golden state in high-dimensional systems. We bridge the gap between the resource theory of superposition and coherence in the context of golden states by establishing a continuous relation by means of a Gram matrix. In addition, immediate corollaries of our framework provide a representation of maximal states which reduces to the maximal state of the coherence in the orthonormal limit of pure basis states.