Equiangular linesThe goal is to attack the following problem:
Problem Find as many 1-spaces in ℂn as possible, such that any two of them make the same angle.
Here "angle" is defined as follows: equip ℂn with the standard hermitian form. Given 1-spaces l, m in ℂn choose unit vectors u in l and v in m and define the inner product between l and m to be the absolute value of the inner product of u and v (which is independent of the choice of u, v), and the angle between l and m to be the arccos of their inner product. A collection of 1-spaces (or "lines", or projective points) as in the problem is called equiangular.
It is not hard to see that n2 is an upper bound on the size of an equiangular set of lines.
Conjecture 1 There exist n2 equiangular lines in ℂn.
Why should we believe this? Because numerically such lines have been
found up to n=45, which up to machine precision are equiangular.
These examples are of the following shape. Let A be the matrix diag(1,z,...,zn–1) where z=exp(2πi/n) and let S be the shift matrix
0 0 0 1
1 0 0 0
0 1 0 0
0 0 1 0
(for n=4). Modulo scalar matrices, A and S commute. Their images in the projective unitary group PUn generate a group PH isomorphic to ℤn x ℤn. (In Un, they generate the Heisenberg group H which has the group of scalar matrices <μI | μn=1> as centre, the quotient by which is ℤn x ℤn.)
Conjecture 2 There exists an equiangular PH-orbit of size n2.
Fact A set of n2 equiangular lines li would maximise ∑i,j <li,lj>4.
This leads to the following numeric approach: optimise the function that on input of a line l computes the latter expression for the PH-orbit of l. Using numerical optimisation, a suitable l ("fiducial state") has been found for n up to 45, thus numerically supporting conjecture 2. See this page.
Low dimensionsConjecture 2 has been verified in dimensions 2,3,...,15,19,24,35,48 in the papers below.
NormaliserThe normaliser of PH in PUn is not hard to compute. Of course it operates on the lines whose orbit is an equiangular set with size n2. Modulo PH the normaliser is isomorphic to SL2(ℤn). It was observed by Zauner that a suitable "fiducial state" is often an eigenvector of an element of order 3 in this normaliser---unfortunately the corresponding eigenvector has multiplicity larger than one.
SIC-POVMs: A new computer study
Numerical SICPOVM solutions
Equiangular tight frames from Paley tournaments
Symmetrically information complete quantum measurements
Some remarks on Heisenberg frames and sets of equiangular lines
On SIC-POVMs and MUBs in dimension 6
Quantum measurements and finite geometry
Joe Renes's dissertation
Markus Grassl's slides
Gerhard Zauner's dissertation