# Equiangular lines

The goal is to attack the following problem:
**Problem**
*Find as many 1-spaces in ℂ ^{n} as possible, such that
any two of them make the same angle.*

Here "angle" is defined as follows: equip ℂ^{n}
with the standard hermitian form. Given 1-spaces *l*, *m*
in ℂ^{n} 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 *n*^{2} is an upper bound
on the size of an equiangular set of lines.

**Conjecture 1**
*There exist n*^{2} *equiangular lines in ℂ ^{n}.*

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*,...,*z*^{n–1}) where
*z*=exp(2π*i*/*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 PU_{n} generate a group PH isomorphic to
ℤ_{n} x ℤ_{n}. (In U_{n}, they generate
the Heisenberg group H which has the group of scalar matrices
<μI | μ^{n}=1> as centre, the quotient by which
is ℤ_{n} x ℤ_{n}.)

**Conjecture 2**
*There exists an equiangular PH-orbit of size n*^{2}.

**Fact**
*A set of n ^{2} equiangular lines l_{i}
would maximise
∑_{i,j} <l_{i},l_{j}>*

^{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 dimensions

Conjecture 2 has been verified in dimensions 2,3,...,15,19,24,35,48 in the papers below.

### Normaliser

The normaliser of PH in PU_{n}is not hard to compute. Of course it operates on the lines whose orbit is an equiangular set with size n

^{2}. Modulo PH the normaliser is isomorphic to SL

_{2}(ℤ

_{n}). 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.

### Some references

SIC-POVMs: A new computer studyNumerical 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