Game of Sloanes: best known packings in complex projective space

Abstract

It is often of interest to identify a given number of points in projective space such that the minimum distance between any two points is as large as possible. Such configurations yield representations of data that are optimally robust to noise and erasures. The minimum distance of an optimal configuration not only depends on the number of points and the dimension of the projective space, but also on whether the space is real or complex. For decades, Neil Sloane's online Table of Grassmannian Packings has been the go-to resource for putatively or provably optimal packings of points in real projective spaces. Using a variety of numerical algorithms, we have created a similar table for complex projective spaces. This paper surveys the relevant literature, explains some of the methods used to generate the table, presents some new putatively optimal packings, and invites the reader to competitively contribute improvements to this table.

Continuous Welch bounds with Applications

“…Theorem 2.20. [7,11,17,20,24,26,30,40] Define m := dim R (K)/2. If {τ j } n j=1 is any collection of unit vectors in K d , then…”

“…There are several practical applications of Theorem 1.1 such as correlations [28], codebooks [14], numerical search algorithms [40,41], quantum measurements [29], coding and communications [31,35], code division multiple access (CDMA) systems [22,23], wireless systems [27], compressed sensing [34], 'game of Sloanes' [20], equiangular tight frames [32], etc.…”

Bounds on Antipodal Spherical Designs with Few Angles

“…It is particularly interesting to consider two special cases, i.e., s = 3, t = 3 and s = 4, t = 5. These two cases are closely related to the optimal line packing problem, which aims to find a finite set Φ = {ϕ i } n i=1 ⊂ S d−1 with fixed size n > d and the minimal coherence µ(Φ) := max i =j | ϕ i , ϕ j | (see [10,13,15,17]). The followings are two well-known lower bounds on the coherence:…”

Quantum state tomography as a numerical optimization problem

“…. , m} in an optimal, maximally spread fashion has been studied for both F = R and F = C [28][29][30][31][32][33][34][35][36][37][38]. Typically, optimality here refers to maximizing the minimum chordal distance d 2 c (P j , P i ) = l − Tr(P † j P i ) where P j is the projector on the subspace U j , i.e., min i =j d 2 c (P j , P i ) shall be maximal.…”

“…We describe the realization of such a system -in NV centers in diamond. We reveal the relation between our optimization problem of finding the optimal QST measurement set and packing problems in Grassmannian manifolds, which have been studied in great detail [28][29][30][31][32][33][34][35][36][37][38] and are relevant for many fields, such as wireless communication, coding theory, and machine learning [37,[39][40][41][42]. As we are able to approximate the optimal measurement scheme of the qubit-qutrit system, we solve a greater problem, namely we find an optimal Grassmannian packing of halfdimensional subspaces in Hilbert space of dimension six.…”