The construction of optimal line packings in real or complex Euclidean spaces has shown to be a tantalizingly difficult task, because it includes the problem of finding maximal sets of equiangular lines. In the regime where equiangular lines are not possible, some optimal packings are known, for example, those achieving the orthoplex bound related to maximal sets of mutually unbiased bases. In this paper, we investigate the packing of subspaces instead of lines and determine the implications of maximality in this context. We leverage the existence of real or complex maximal mutually unbiased bases with a combinatorial design strategy in order to find optimal subspace packings that achieve the orthoplex bound. We also show that maximal sets of mutually unbiased bases convert between coordinate projections associated with certain balanced incomplete block designs and Grassmannian 2-designs. Examples of maximal orthoplectic fusion frames already appeared in the works by Shor, Sloane and by Zauner. They are realized in dimensions that are a power of four in the real case or a power of two in the complex case. B. G. B. was supported in part by NSF DMS 1412524, J. I. H. by NSF ATD 1321779. 1 Proof. The cardinality requirement in the definition of orthoplex-bound achieving fusion frames is satisfied since n > d F (m) + 1. Let k, k ′ ∈ K. If k = k ′ , then tr P (k) J P (k ′ ) J ′ = l 2 mfor every J , J ′ ∈ S by Proposition 3.7. If k = k ′ , then the fact that S is l 2 /m-cohesive yieldswhich shows that F is an orthoplex-bound achieving fusion frame. Finally, if S is maximally orthoplectic and the set of mutually unbiased bases is maximal, then Corollary 3.9 shows that the coordinate projections belonging to each basis sum to a multiple of the identity, so the corresponding fusion frame is tight. Hence, the union of all the coordinate projections belonging to the mutually unbiased bases forms a set of n = k F (m) (2m − 2) = 2d F (m) orthogonal projections whose pairwise inner products are bounded by l 2 /m, which shows that F is a maximal orthoplectic fusion frame.Following Zauner's ideas, we repeat the study of design properties for the special case of a fusion frame formed by coordinate projections. To this end, we define the diagonal coherence tensor,where J is the set of all subsets of [[m]] of size l, and, for each J ∈ J, D J is the J -coordinate projection with respect to the canonical basis. An elementary counting argument shows D 1,l,m =

Equiangular tight frames are examples of Grassmannian line packings for a Hilbert space. More specifically, according to a bound by Welch, they are minimizers for the maximal magnitude occurring among the inner products of all pairs of vectors in a unit-norm frame. This paper is dedicated to packings in the regime in which the number of frame vectors precludes the existence of equiangular frames. The orthoplex bound then serves as an alternative to infer a geometric structure of optimal designs. We construct frames of unit-norm vectors in K-dimensional complex Hilbert spaces that achieve the orthoplex bound. When K − 1 is a prime power, we obtain a tight frame with K 2 + 1 vectors and when K is a prime power, with K 2 + K − 1 vectors. In addition, we show that these frames form weighted complex projective 2-designs that are useful additions to maximal equiangular tight frames and maximal sets of mutually unbiased bases in quantum state tomography. Our construction is based on Singer's family of difference sets and the related concept of relative difference sets.

This paper concerns the geometric structure of optimizers for frame potentials. We consider finite, real or complex frames and rotation or unitarily invariant potentials, and mostly specialize to Parseval frames, meaning the frame potential to be optimized is a function on the manifold of Gram matrices belonging to finite Parseval frames. Next to the known classes of equal-norm and equiangular Parseval frames, we introduce equidistributed Parseval frames, which are more general than the equiangular type but have more structure than equal-norm ones. We also provide examples where this class coincides with that of Grassmannian frames, the minimizers for the maximal magnitude among inner products between frame vectors. These different types of frames are characterized in relation to the optimization of frame potentials. Based on results by Łojasiewicz, we show that the gradient descent for a real analytic frame potential on the manifold of Gram matrices belonging to Parseval frames always converges to a critical point. We then derive geometric structures associated with the critical points of different choices of frame potentials. The optimal frames for families of such potentials are thus shown to be equal-norm, or additionally equipartitioned, or even equidistributed.

In this survey, we relate frame theory and quantum information theory, focusing on quantum 2-designs. These are arrangements of weighted subspaces which are in a specific sense optimal for quantum state tomography. After a brief introduction, we discuss the role of POVMs in quantum theory, developing the importance of quantum 2-designs. In the final section, we collect many if not most known examples of quantum-2 designs to date.

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