Equiangular lines

Equiangular lines in Euclidean space

Computing the maximum number of equiangular lines in n-dimensional Euclidean space is a difficult problem, and unsolved in general, though bounds are known. The maximal number of equiangular lines in 2-dimensional Euclidean space is 3: we can take the lines through opposite vertices of a regular hexagon, each at an angle 120 degrees from the other two. The maximum in 3 dimensions is 6: we can take lines through opposite vertices of an icosahedron. It is known that the maximum number in any dimension n is less than or equal to ( n + 1 2 ) . The maximum in dimensions 1 through 13 is listed in The On-Line Encyclopedia of Integer Sequences as follows:

1, 3, 6, 6, 10, 16, 28, 28, 28, 28, 28, 28, 28, ... (sequence A002853 in the OEIS)

In particular, the maximum number of equiangular lines in 7 dimensions is 28. We can obtain these lines as follows. Take the vector (-3,-3,1,1,1,1,1,1) in R 8 , and form all 28 vectors obtained by permuting the components of this. The dot product of two of these vectors is 8 if both have a component 3 in the same place or -8 otherwise. Thus, the lines through the origin containing these vectors are equiangular. Moreoever, all 28 vectors are orthogonal to the vector (1,1,1,1,1,1,1,1) in R 8 , so they lie in a 7-dimensional space. In fact, these 28 vectors and their negatives are, up to rotation and dilation, the 56 vertices of the 3₂₁ polytope. In other words, they are the weight vectors of the 56-dimensional representation of the Lie group E₇.

Equiangular lines are equivalent to two-graphs. Given a set of equiangular lines, let c be the cosine of the common angle. We assume that the angle is not 90°, since that case is trivial (i.e., not interesting, because the lines are just coordinate axes); thus, c is nonzero. We may move the lines so they all pass through the origin of coordinates. Choose one unit vector in each line. Form the matrix M of inner products. This matrix has 1 on the diagonal and ±c everywhere else, and it is symmetric. Subtracting the identity matrix I and dividing by c, we have a symmetric matrix with zero diagonal and ±1 off the diagonal. This is the Seidel adjacency matrix of a two-graph. Conversely, every two-graph can be represented as a set of equiangular lines.

Equiangular lines in Complex vector space

In a complex vector space equipped with an inner product, we can define the angle between unit vectors a ^ and b ^ by the relation cos ⁡ ( θ ) = | ( a ^ , b ^ ) | . It is known that the maximum number of complex equiangular lines in any dimension n is n 2 . Unlike the real case described above, it is possible that this bound is attained in every dimension n . The conjecture that this holds true was proposed by Zauner and verified analytically or numerically up to n = 67 by Scott and Grassl. A maximal set of complex equiangular lines is also known as a SIC or SIC-POVM.